Duma's post is an awesome place to start out if you're not familiar with how to prep and battle. And Gypsyknees has some great tips! Jaydeed has a good point too. If you are looking for certain prizes, like stamps or books, etc. TDN's pages on the 1p challengers has prize data too! And boy did it take a long time to collect! lol
If you post your battle pet stats, and available equipment if you have some already, people can also help you out with different builds (ways to train your pet) for a style that works for you.
Just don't go into battle with nothing equipped! LOL When I first joined Neopets, many years ago, I figured there would be a default action to attack without any equipment. there wasn't. lol This was in the old battledome of course, but still, go in with something, even against Punchbag Bob, or you'll have to withdraw, which will count as a loss. That was my experience anyway. lol
As for opponents. Before the prices dropped so much, I liked to battle the Jelly Chia. He drops some stamps and a book. Jelly Pop Stamp, Jelly World Stamp, and Jelly Mysteries which is a book. Made some nice NP on the stamps back when I was trying to figure out all of Jelly Chia's prizes. Not sure what they sell for now, but the last time i checked they were both still very much under 100k np on the shop wiz.
Different domes lend to different prize types too. If you take place in the Obelisk Skirmishes, those are all in the Ugga Ugga Dome. The prizes as I recall from there are so common in the user market because of this. There might be a few decent rare prizes there. I usually battle some other opponent in a different dome before I do the battles for the skirmishes. Rattling Cauldron Dome, basically the Haunted Woods Arena, you'll get lots of spooky foods. A lot of which is not as valuable as it once was because they are common prizes in that arena now. Neopia Central Dome has a lot of regular food items. If you're lucky you might get Chocolate Ice Cream, worth about 40k the last time I won any. lol Again each opponent also has some pretty unique stuff, as well as codestones of course. Shadow Usul, for example, has a plushie that sometimes drops, and sells for around 15k-20k.
Anyway, depending on your budget, stats, and the opponent(s) you'd like to battle, post that info either here or in a new thread, and I'm sure you'll get lots of tips. I spent my time budget battling for as long as I've been on Neopets. I'd say that my battle set is still in the low-intermediate to regular intermediate range. I've a few weapons from plots I earned. The majority of my NP spent for battle gear was on my healer, and a Ghostkerbomb. So I consider budget-battling my specialty.
I recommend training over expensive weaponry. I mean, I've got my Strength and Defense stats maxed, but my weaponry is not terribly great, as I said above. If you read about "Boosts", those are basically the multipliers for the amount of icons a weapon attacks or defends. If you have a 10 icon weapon, and you've got the highest strength boost, 16, which you get when your strength is 750. Then that attack, if nothing is blocked by your opponent, will do 160 damage.
I also recommend balanced training between Strength and Defense. But that's my training style. lol Movement/Agility is pretty useless except for a couple of items. If you have agility of 201 and an intelligence of... I believe it was 37+, you could consider a Heavy Robe of Thievery to steal a random item once per battle, but stealing is not working right now, so the point is moot right now. Though it might be a good time to buy a Heavy Robe of Thievery for the future, because when stealing gets fixed, the price will probably go back up a bit.
Intelligence, it's useful for some higher-end tunics from the Brightvale Armory. But reading books and visiting Coltzan's Shrine should yield you some decent INT increases. I think the most Intelligence you need for a pet in the BD would be 1000 for one of the tunics. Again though, not something to worry about if you're just starting out. The USEFUL tunics sell for some millions mostly.
Phew! I haven't pulled a post this long in awhile. Sorry, i hope it all makes sense and wasn't too much at one time.
Happy battling!