My advice based on what you have said, and basically my advice to everyone is a heifer. Unless you are only showing at a smaller not super competitive county show that has a sale, then maybe you can make a steer work. But the steer shows can be brutal, it can be super expensive to get one ready, and then you have to make the sale and you are basically stuck if he doesnt make the sale. If you buy a heifer that is quality enough to show then you will at least have something to sell as a replacement and not totally lose everything on.
As far as price, yes I believe if you are going to have something to be show quality then you are going to spend at least $1,000 if not $1,500 at a minimum. But there are always some steals out there, my advice is always talk to some people, find someone you trust that has a background in the industry that is good at prospecting calves and shop around. I can typically find a bunch of really good buys on really good cattle every year because they are green and people simply dont know how to prospect them out. Or they think that they wont grow hair when the calf has been on pasture in 100 degree heat and if you put it in a cooler like the big fluffy pretty ones in a lot of these sales it will have as much if not more hair.
As far as breed, your gonna have every breed in the book thrown out there, because there are breeders of each breed that are partial. Personally I can see the value and purpose for just about every breed. I think in choosing a breed you have to stop and consider what your ultimate goal is, and find the breed that best fits it. If your ultimate goal is show cattle and you dont have a breed preference, then I think the best thing for you to do(especially on a budget) is throw breed out the window and go out there and find the best calf you can possibly find in your price range and buy it, regardless of breed. If you are thinking you ultimately want to produce show steers, then I would probably swing a little more in the direction of an exotic type breed. Shorthorn Plus, MaineTainer, Chi, etc. Your more likely to find the phenotype you want in those breeds if that is what your looking to do, plus there are a lot more breeding options out there. I know I will probably make a lot of people mad out there by saying this, and trust me I have nothing against any of these breeds. But if I were on a budget there are a few breeds I tend to stay away from Angus(very very competitive breed shows, can be kinda cut throat, the ones that win are gonna have a ton of money behind them, and my personal experience is they are one of the most political breeds out there.) If your in the south, the Brahmans can be the same way. Not saying that the blind squirrel doesnt sneak in those breeds and get the nut sometimes, but its not typically how it goes.
My advice, heifer has more options and opportunity for you in the long run, you can show for 2 yrs instead of just one. As for breed, unless you are partial to one then find the best calf you can for the money you can spend and dont worry about what breed it is, just make sure you have shows for those breeds in your area. Lastly, shop around, dont go buy the first pretty calf you find that is in your price range, and dont just buy a calf because it is in your price range. Even if a calf is under your price range, if it is a complete train wreck within your budget, then you are gonna lose it in the nd anyway,, and you wont be happy or enjoy what you are doing.