Yay! You've decided to be a Battledomer on Neopets! Congratulations! Anyways, let's just get to the point, shall we? XD
Anyways, the first major tip to being in the Battledome is buying multiple use items - items that aren't snowballs, etc. One-use items are good, but you can only use them for one turn. Also, make sure your pet has full hit points before entering.
But before that, you'd probably want to train. I'd suggest that you buy some codestones and train in strength, defense and endurance. Those are the things that will help you in a fight. Agility and all the other stuff won't help - it doesn't do you much good in a fight. If buying codestones every time is not appealing to you, try buying all ten pieces of the Secret Laboratory Map to get the Lab Ray. :)
Now, you've trained your pet and you got it to full hit points. What next? Now, we go on to the expensive part - buying weapons. If you're a newbie, your financial resources on Neopia are bound to be limited - you're most likely not a millionaire. Therefore, items are most likely to be one-use and not multiple-use. Here's a list of sets that I recommend now:
Not much NP Set:
x6 Sticky/Exploding/Icy Snowballs
1 Attack Fork
1 Slime Potion/Greater or Lesser Healing Scroll
Now, if you have a lot of NP (at least 100K - 500K or above), I suggest you get this set:
Lot of NP Set:
1 Bronze Scorchstone/Greater Healing Scroll
1 Virtupets X-514 Super Shield
2 Staff of Brain
1 Black Frost Cannon/Yellow Sticky Hand/Freezing Potion
2 Mud Mixtures
1 Caustic Potion/Honey Potion
You got Healing, Defense and Attack all in one package. With this package and a properly trained pet, you can probably take on a challenger with at least 20 difficulty.
Let's move on to your first fight! The easiest opponent would be by far Inflatable Balthazar because he has 1 HP and he doesn't fight back. There, you can practice how to fight so you know how when you face against a human or something.
Eventually, you'll get better - practice makes perfect, eh? The Battledome is really fun, you know. :) Anyways, just to let you know, healing scrolls have a finite life time, which means they die after some uses - I think it's about 25 - 50 uses, so be careful.
Finally, Tooth of Terask is a decent multiple-use attack item - only 50K for one. You can get one from beating NeoQuest II on Evil mode.