I have personally tried Premiere and I think it sucks. "Professional" apps like Avid and Final Cut Pro will give you less headaches. A big problem with Premiere is that it lets you lose sync between audio and video and that it takes more keystrokes and clicks to do a task then other programs. However, the people at my school have found it easy enough to use (the import/export settings are a bit daunting though). Premiere isn't used professionally because of the hassles it gives. Things might change with Premiere Pro. I hear good things about Vegas Video, but I haven't tried it myself (although I am planning to once I get some $$$). Avid also might be a choice too, and if you get Automatic Duck it integrates well with After Effects (but that's very costly).

If I had $3k-$4k for a computer, I would definitely get a Mac unless I needed to play games or run certain PC programs. With educational pricing, you could get a G5 (insert drool here). But anyways, you want a PC and premier and After Effects.

Some companies sell turnkey systems for Premier and Vegas Video (and I think Edition too). For Premiere you can get RT effects cards that speed it up (although they may not work with premiere pro if you buy em now). I know Promax sells turnkey systems, and other companies probably do too. If going with Premiere Pro, get a Pentium based system as PP will be optimized for it.

If you want great savings, build your own system. For advice on components to get, go over to the ars technica forum and ask for help (read the stickies first). There are resources on the internet on how to put together a computer. You can get a hot rod computer for $2k. Add on a RT effects and FW card and you'll be set. If you are buying Premiere now, it's cheaper when you buy certain capture cards.

http://arstechnica.com/guide/system/hotrod.html
There's a buyer's guide for what to get, but it implies overclocking. Total system cost is $1200 + shipping + OS. For hundreds more get the fastest CPU and the right motherboard if you don't want to overclock. The fan will be excessive since the retail fan is good enough. Dell is dull, build your own computer!

However, if you wanted a cool computer, check out ocsystem.com They sell stable, overclocked computers (faster than Dull) which look damn purty. Maybe even better looking than the G5s eh ccbatson?