The Altimas are great cars, but I drove one for awhile (2002 with the qr25) and its handling was terrible. It was even slightly modified with I/H/E and had KYB agx shocks and H&R sport springs. It's more of a cruiser/luxury car and it drives like one.
If you want to get into modifying cars then a good place to start is a 92-95 Civic. They are very cheap, have TONS of potential, are soooooo easy to work on, and have a gigantic aftermarket base.
Or perhaps a 94-95 integra. I advise staying OBD-1 because if you start modifying it OBD-1 is just easier to deal with. I can also give you a few pointers, because not only did I have my own modified civic for a few years, but I've been a moderator over at superhonda.com for several years and also I'm a motorsport engineering major
I'm a Honda fan so will have a tough time recommending other things...Hondas do have their drawbacks like most being FWD and pretty much all of them not having much torque at all, but as a first car and intro into modifying stuff they are ideal...and when built correctly they can be wicked fast.
On the other hand like Amd_gt_player said, the 240sx is also a great car. It's RWD but is not turbo and feels pretty slow at stock, so you won't be in much danger of spinning out or losing control. You can slowly modify it as you get used to the car, they have great potential also. Lots of people swap in the sr20 or ca18 but I think that's a waste, you can just boost the stock ka24de and have a better powerband.