Since you have $800..
Stick with the E6300 CPU - it'll be a great processor for a long time and is relatively inexpensive.
Since you're not into overclocking but do a little gaming, a wide array of motherboards are available for varying prices. Do you want to eventually be able to upgrade to nVidia SLI? (the x300 is rather low-end right now anyway, but will work for now) If so, look at the Asus P5NSLI for $107. For something newer, go with the Gigabyte 965P-DS3 ($130) or the Asus P5B Deluxe ($177).
For RAM, (1GB) 2x512MB DDR2-800 should be a minimum.. if you're planning on Windows Vista in a couple months, just get 2GB now. CorsairXMS DDR2-800C5 is around $250 for 2GB (2x1GB).
For cases, the only one I'd seriously consider is the Antec P180 - the best combination of price, silence, and cooling. Some people don't think it's flashy enough, but I like the sleek style. Either in black or two-tone silver/black for around $130-140 shipped.
For power supplies, SeaSonic are very quiet, efficient, and stable. The 430W vesion is $90 and more than enough power.
There are a few great 19" LCD monitors (widescreen and standard aspect) for around $200 by ViewSonic.. you could get an Antec case w/350W power supply for around $60 shipped, the E6300 ($181), the P5NSLI ($107), 1GB Corsair XMS2 DDR2-800 ($125), Western Digital SE16 250GB hard drive ($80), Lite-On DVDR burner ($30).. then get the ViewSonic VA1912wb 19" widescreen LCD ($228) with DVI and VGA inputs. (Most of the other 19" models don't have a DVI input.. spend the extra $30 for this model)
That should be around $800-820... if you need to cut corners, re-use a current hard drive for awhile until you've got some extra money for that upgrade. Since you're not into overclocking, you don't need high-end RAM. With the E6300's 1066MHz bus, you don't technically need anything higher than DDR2-533, and could go with the Corsair ValueSelect 2x512MB for $107.