Well, lets get your definition of good handling. I'd say something better than your pickup and at least as good as a modern sedan? Fast and quick can be qualified on a drag strip, but handling and ride quality are pretty subjective and can mean different things to different people. We can get your car to feel like a true performance vehicle, but if you think a Vette or Viper ride too harshly, then you certainly don't want your E body to be similar. I also figure you don't want old school Cadillac mush either.
Also, what do you want to do for budget? $100, $500, $1000, $10,000?
How much can you do yourself? Got basic hand tools or a decent sized box? Do you have a service manual? Can you remove/press bushings or are you farming that out?
Cheap, well that is somewhat relative based on budget. Cheap shocks for a $500 build are a whole lot different than cheap shocks for a $10,000 build.
1st things 1st and probably the cheapest change you can make; dump the OEM alignment specs and get something more in the realm of modern specifications. Set the ride height where you want it, tell the shop not to alter it and ask them for; Caster-as much positive as your shop can get while balanced against negative .5 camber and 1/16 inch toe in. This should make a very noticeable difference over the stock alignment specs. Cheap fix here for arond $50 depending on shop rates in your area. Another cheap item, Moog offset upper control arm bushings for $50 to get more caster.
Shocks, well, good shocks can make a otherwise stock set up feel really good. However, bad shocks can make an excellent collection of parts feel worse than lousy. Shocks are to the suspension what cams are to engines; the are the core of it all. Some of the best shocks for all around conditions that allow you to test and experiment with producing the ride and feel you want are QA1. They are bolt in, they are adjustable, and they are under $150 each. These will work with whatever sized components you have under there and can grow with whatever changes you want to add in the future. But, again, budget may dictate otherwise of you want to look at a whole package. Stock or slightly larger than stock set up can use Edelbrocks around $80 each. KYBs at $45 are decent if you car weigh upwards of 4000# or has worn out components. I don't care for them much and would take off the shelf Gabriels over them.
What size are the sway bars, t-bars, and leaf count? Knowing where you are now will help us determine where you need to go. Other items such as engien family, steel or fiberglass body components, etc...