Some great places to get free stuff are:

Tombola
The Fruit Machine
Coltzan's Shrine
The Giant Omelette
The Giant Jelly (I'd give the link, but Neopets has blocked it.)
The Wheel of Excitement (Not free, but still good)
The Healing Spring
Grumpy Old King
The Snowager (Visit him at 2pm and 10pm during naptime)

You could also try the Rubbish Dump or the Money Tree, but I really see no point in either. Too much competition, and the items that you can typically get there are not the best for selling.

2. Play as many games as you can until you can find one that you can earn the max (1000np per game) three times a day. Remember, if one is good then two is better! Personally, I play Meepit Juice Break and can easily earn 3000 from it, although this might not be the game for you. If you don't know where to start, try playing the Featured Game every day.

3. Learn the Cellblock strategy. What is Cellblock you ask? Why it's a game that you can earn a max of 5000 np each day. Not only that, it's a breeze to do so! There is a fool-proof strategy at Pink Poogle Toy. You'll have to do a search on Google or some such search engine because neopets does not allow people to link directly to outside sites. Or give this one a try at Seraphina222's petpage made by sarahtheo.

4. Now that you can reliably earn at least 8000 np per day, time to start in on Bilge Dice. This is a game of luck that really pays off big. Unfortunatly for it do so, you need some serious patience. Since the rules are explained on the Neopets site, I'll just tell you how to parlay that into 7900 neopoints a day.
First bet as much as you can. In the beginning this will only be 50 np. After you win a certain number of games you'll get higher antes. The goal is to eventually get the 1000 np one. Just keep playing until you do, and don't worry: if you win at least a third of the time you'll more than break even.
Once you have the 1000 np ante, try to get your total won up to as close to 5000 as you can without going over. Why? Once you go over 5000 you are done for the day. If you can get it up to 4900 and then bet 1000 and win, you'll get to keep the overshoot which will amount 7900 np total. Pretty sweet, don't you think?
Of course since this is a gambling game there is a downside. You might end up a few thousand neopoints in the hole. Just don't panic! If you keep playing you will eventually drag yourself out again. Have patience and keep at it and you will get your 7900 np.

5. Seeing as how you're earning some decent neopoints from games, time to start focusing more on your shop. To get some good items to sell in there, hang out in the Neopia Central Shops. You can buy items, and mark them up to sell in your shop. Shops fill up randomly, but they always do at exactly :00 on the second hand of a clock, and sometimes again at :08 (or maybe it's :10. Close enough. :P). So if you refresh the page at :00 (or just after that) you'll see the items as they appear, and are more likely to get good ones.

Some of the better places to buy from are:

Chocolate Factory
Book Shop
Bakery

Nearly everything bought in these stores can be marked up for a profit. Overtime you'll learn the best items to grab and can increase your profits accordingly. That said, avoid Usukiland, the Pharmacy, Defense Magic, Attack stuff, and the Magic Shop like the plague until you learn what items can be bought for a profit. There are many sold in these shops that you'll have to resell at a loss, sometimes of thousands of np. (Just because an item is expensive, doesn't mean it is good!)

Also, the 3rd of every month is Half-Price day, where everything in the Neopets run shops is half-off! (The user shops don't change though, and neither does the Hidden Tower.)

It is harder to restock if you have dial-up rather than a fast connection, but just be diligent. You can always get something, even if it's not the best item in there.

6. Once you can reliably earn 15,000 neopoints a day or have at least 100,000 np in the bank it would be a good idea to start investing in the Neopian stock market. The stock market is the easiest way to earn lots and lots of neopoints. What other game only takes 5 minutes a day and is a near fool-proof way to make millions? I love the stock market! I love is almost as much as my guild!

Seeing as how the NeoDaq is fairly complicated to a newbie, I will direct you to Kuraddo's petpage, where an excellent stock market guide is located.

Follow all these steps and you'll be a neopian millionaire within a few months, maybe a year at most! Don't forget to experiment with your own money-making ideas; the sky is the limit.

But before I go, I'd like to say a special thank you to AirMystic, apoxy, Crudmonger, nova_442, jaban0, Auntieent, my real-life little sister kembryan, all the members of Adult FUNdamentals guild, and the regs on the Stock Tip board. Without their advice and tips and feedback it would have taken me a lot longer to achieve a nice healthy bank balance.