Act I

The first thing you should do is immediately exit The Rogue Encampment into The Blood Moor. There's simply nothing productive to do in town yet.

Before you start killing monsters, open the chat and type the following command: /Players 8
An automated message that says "Players set to 8" should appear.
What this does is adjusts the game difficulty from 1 to 8 (8 is the highest). Adjusting the game difficulty increases three things.
-The amount of HP monsters have
-The amount of experience you get from monsters
-The amount of loot that monsters drop

Naturally, I am suggesting that you play on the highest difficulty from the start, because the experience and loot are well worth it. At times I will have you turn it back down to 1 and then later on back to 8 again. Basically, any time you merely need to get from Point A to Point B as fast as possible, you should be on the lowest difficulty, because it being higher only makes things take longer.

Be sure to change difficulty in town before you enter the next area. The difficulty of an area is set when you enter it. So if you enter Stony Field while difficulty is set to 8, the nearby portion of it will be stuck on 8, even if you try to change it back to 1. The only way to reset that is to leave the game and enter it again.

Whether or not you enter The Den of Evil right away and clear it out for the skill point is up to you, but I usually wait until my characters' build can do it more efficiently later on. After all, a level 2 or 3 typically doesn't benefit much from an extra point anyway.

Kill monsters in the Blood Moor for a bit. Pay close attention to what drops, because if you're lucky, some of the items that sell for lots of gold (For this point of the game anyway) may drop. You want to buy a Scepter and good armor as soon as possible, so the sooner you have the gold for it the better.

The following items are occasionally dropped in the Blood Moor and sell for a lot of gold:

-Amulets & Rings:
800-2,500 Gold

-Staves, Wands, and Scepters that add class skills:
200-2,500 Gold

-Throwing Knives:
 480 Gold  

-Rejuvination Potions:
200 Gold

-Rare items in general:
50-1,500 Gold

Also keep an eye out for gems, but keep them in your stash. DO NOT socket your armor with rubies, sapphires, or topaz yet. You will need these for weapons first.

Why a Scepter? Regardless of whether your character is a caster or not, it is going to need to lean into being a melee character until its skills mature. Particularly, a Sorceress can't effectively rely solely on her skills until at least level 14, while a Summon Necromancer may only need to melee until level 6, when it its skeletons can reliably do all of the work.

The Scepter is the best DPS weapon you can get until the Flail in Act II. While there are weapons with better damage than the scepter, they are either two handed or have slow attack speed. The weapons with faster attack speed have a lot less damage. The scepter is a nice balance of damage and attack speed.

Also, never use two-handed weapons, at least not at the low levels. Shields are invaluable and your characters block percentage is important. While I don't recommend pouring points into dexterity this early for block for multiple reasons, melee characters will need 35 Dexterity to use a Flail when they get to Act II.

As far as stats go, your first focus should be to hit 40 base strength as soon as possible to wear better gear. For melee characters you may take this further to 55. After that, it's up to you whether you want a little more block percentage or more life.

I caution that you shouldn't add too much to dexterity too early though, as you will get shields with better block (Particularly the 30/20 Bone Shield I will have you shop in Act II) that will get more out of that dexterity. The reason that may be an issue is because you may have more dexterity than you need when you upgrade your shield. For instance, let's say your paladin needs 90 dexterity to have max block with a Large Shield, which has 42% block. So you foolishly add dexterity until you have 75% block with it. You then get a 30/20 Bone Shield, which has 70% block. You would have only needed to have 70 dexterity to have max block with that bone shield, so you have 20 points that are currently a waste.

 When you have items that you are confident will net you at least 600 Gold, immediately head back to town and sell them, then shop at Akara for a scepter. Now comes the part that some consider boring. If you're impatient, just settle for the first scepter you can afford, but with some patience, you should wait on a 2 socket scepter. If Akara doesn't have what you seek, either walk into The Blood Moor or take a waypoint somewhere (Whichever is closer), then come back and her list of items will refresh. This process is referred to as "Shopping". This process should produce a
2 socket scepter within 5 minutes, but it's very random, so it could take longer.

When you have/get them, you should put one sapphire (To slow monsters with cold) and one ruby or topaz in your Scepter.

If you have some gold left after that, consider getting some of the following from Charsie as well:
-A shield with better block (Large Shield is the best in Act I)
-Studded Leather Armor (2 socket is nice)
-A Skull Cap (2 socket is nice)
-Gloves, Belt, or Boots

Then head back out. If you are still under level 3, stay in The Blood Moor until then. Otherwise, head for Cold Plains. Kill in Cold Plains until level 5.
Then head for Stony Field.

While killing in Stony Field, you will need to find the waypoint and the Cairn Stones. After that, you should get to level 8. Once you are level 8, you should go kill Blood Raven. Get your mercenary (Make sure you get a fire one, you can get cold slow from a sapphire) and when you have enough gold, get her armor, helm, and shop at Charsie for her a 3 socket Hunters Bow.

When have/get them, put one sapphire and any combination of rubies or topaz in the Hunters Bow.

Turn the difficulty back to 1 by typing /Players 1. Then make your way to Dark Wood. Find the Tree of Inifuss and the waypoint. Then go open the portal to Tristram. Now you can set the difficulty back to 8 and do Trist Runs.

Trist Runs are when you clear out Tristram, exit the game, then clear it out again until you are level 17. At the point that you hit level 13, making Nadir (Nef Tir) out of a 2 socket Skull Cap isn't a bad idea.

Once you are level 17, turn the difficulty back to 1.
If your character is a caster, now is a good time (If you haven't already) to drop the scepter and shop a wand from Akara. You are really only looking for two things on it, 10% Faster Cast Rate and mana.

Then head to Black Marsh and find the waypoint and The Forgotten Tower. Farm The Countess for awhile until you get Tal and Eth runes. Having Nef, Tir ,  & El runes are a bonus. If your character is melee, you will need to keep a set of Tir & El to the side for later (If you get them, if not, you can farm for them later when you need them).

When you have Tal & Eth, shop a 2 socket Ringmail from Charsie. This could take 2 minutes or 20 minutes, but this is not optional. You are going to make the Stealth runeword and it's a very powerful armor for the early game. Opting out of it is a bad idea. Make Stealth in it by putting Tal in first, then Eth. This armor will contribute 6 Dexterity to your block, make you run faster, give you better hit recovery, and if your character is a caster, make you cast faster. Even for non-caster characters, the 6 dex, FHR, and faster run can't be beat.

Now make a bee-line for Catacombs level 2 and find the waypoint. When you get to the Catacombs, you could turn the difficulty back to 8 and level up to 20 if you like. It's not necessary but I, for one, like to over-level. Just be sure to turn the difficulty back to 1 before heading to Andariel.

Okay, before you rush to kill Andariel, I want to explain a way to farm her for items. It involves a bug. See, when you kill a boss for the first time, you get a different quality of loot. But there is a way to ensure that Andariel has that quality of loot every time you kill her.

You do it like this:
When Andariel dies the first time, DO NOT open your own Town  Portal. One will appear automatically near her body. Take that one and talk to NO ONE ELSE but Warriv. Have Warriv take you to Lut Golein (Act II). As soon as you spawn in Act II, exit the game.
That's it. From now on, Normal Difficulty Andariel will have better loot than usual. She will drop more rares and have much better chances of dropping set and unique items.

Once this is done, if Andariel wasn't too much of a challenge, I recommend farming her for items for awhile.

                                   Act II

The beginning of Act II is exciting, because the items Drognan and Fara offer provide many gear upgrades, as well as being able to upgrade to an Act II aura mercenary.

First things first, hopefully you have at least 20,000 gold to shop a 30% Faster Block Rate/20% Chance to Block Bone Shield from Drognan. [Bone Shield of Deflecting]. This has the second highest block % for shields in the game. The Tower Shield variants have 3% more, but the strength requirement is too high.

If your character is melee, immediately shop a 2 socket Flail from Fara. If you have Tir & El, make the Steel runeword out of it. If not, go run Countess some more until you do. Steel Flail is an incredible weapon at this point of the game. Certain builds (Like a Holy Fire Zealer) can use it up to level 40 even.

If your character is a caster, the only weapon upgrade you can get is a 20% FCR Wand from Drognan.

Regardless of build, if you have the gold, shopping a 2 socket Splint Mail from Fara to make a better Stealth isn't a bad idea. The same for upgrading to a Full Helm Nadir.

Lastly, get a Combat (Prayer Aura) Mercenary. You can shop a 3 socket Spetum from Fara and make Malice out of it (Ith El Eth) or you can settle for one with at least 25% enhanced damage, since Malice isn't particularly good. However, once you get an Amn rune, consider making Strength (Amn Tir) out of something for your mercenary.

There aren't really any places to farm in Act II until the very end, not for experience or items. So just keep the difficulty on 1 until you reach Canyon of the Magi. Then, put it on 8 and clear out tombs until you are level 22. Put it back on 1 when it's time to kill Duriel. 

                                             Acts III, IV, V

There really isn't much for the rest of Normal Difficulty to do except progress towards Nightmare Difficulty. You may find mercenary weapon upgrades like Pikes & Partizans, but other than that, there isn't much upgrading to do until Nightmare, where Exceptional Items start to drop often.