Actually I think you will be better of with a Dell, unless you really, really want to learn about hardware (I take it you don't) or if there's a specific unusual hardware component you really want, such as SCSI-drives or gigabit ethernet or other oddities. A couple of years ago you could really save money building the computer yourself, but nowadays it's rarely worth it as the Dell prices are mostly excellent for "normal" computers (with normal computer I mean lots of storage, a decent video card, lots of MHz and a TFT-monitor).
But. If you do want to build the computer yourself, it can be quite rewarding. The process is pretty much like this:
1) Insert CPU and heatsink/fan in motherboard. Nowadays this is really simple, basically "plug and play".
2) Insert memory in motherboard. This is the trickiest step as it involves force and violence.

3) Insert motherboard in case.
4) Plug in a couple of cables from the case to the motherboard, such as the power on and reset button.
5) Insert videocard.
6) Install the hard drive by attaching one cable to the hard drive and the other end to the motherboard.
7) Insert the cabels from the power supply unit
8) Almost done (a few minor things left).
I don't know what kind of performance you want from your new computer, but here's an example of a really fine machine. Prices from newegg.com
Athlon64 3000+ (socket 939) = $146.00. I think the Athlon64 is a better choice than the new Pentiums because they are really easy to install and they don't run as hot as Intels new processors.
Abit AV8 motherboard = $109.00. Personally I prefer Abit over ASUS. I know others prefer ASUS though. YMMV.

2x Mushkin 512MB PC3200 RAM = $136.25. A gigabyte of RAM is neat.
Hard Drive: Anything will do. Say 160GB = $100.00
Antec Sonata = $105.00. Great looking chassi with great gower supply.
Thats about $600, add video card and monitor and you are basically set.

You can get it cheaper if you don't need all the power.
Good luck.