For Calc 1, 2 you don't NEED to have a calculator capable of symbolic calculus functions. Helpful sometimes, sure, but most people get through with nothing more than a TI-83+, which definately doesn't do symbolic integration.
For calc 3 you also don't NEED a calculator capable of 3D graphing. The 3D graphing is much more useful for calc 3 than symbolic integration and the like is for calc 1 and 2, however.
So, if you're not worried about 3D graphing, I'd recommend a lower-end TI calculator, such as the 83/84, or, even better, the HP 39G+, which can be had for $65, very close to your budget! However, to be honest, I think you'd want to get something a little more high-end.
So, I suggest that if you want 3D graphing and symbolic calculus capability, you should pick up the HP 48GII for $85. This calculator also supports RPN, if that interests you at all. The main differences between it and the higher-end 49G+ is mostly that the latter has more memory, a bigger display, and a faster processor. It's really quite a steal. For comparison, the cheapest I've seen the original TI-89 for is $110.
It will do everything you want it to do, and it is by far the best deal out there. I have a 49G+ and I love the thing to death.
For an interesting article about calculators, see here:
http://www.rskey.org/buyguide.pdf