The best thing to do would be to do what most of us did when we first got our phones and just browsed around. This is a good place to start; there's also Modaco and XDA-developers that have some good stuff. As suggested, flashing a new ROM will make the phone faster and more responsive, plus you'll have the latest cutting edge build of Windows 6.5 (better than what was just released a couple of weeks ago). 6.5 will get you the Windows Marketplace, which is a nice gateway to new applications. Looking at my application list now, I've got Midomi (song identifier), AP Mobile (news), YouTube (obvious), Facebook, BeejiveIM (instant messaging), MobileRemote (connects to PC via Bluetooth and turns the phone into a wireless keyboard/mouse), TCPMP Phenom 3 (awesome video player, plays just about any video file), FVMPlus (free "visual voicemail" type program), Tellme (voice activated web searches/dialing/text messaging), and Pandora (custom streaming audio). I almost forgot games - SPB Quads, SPB Brain Evolution 2 and SPB Geo Game (look them up on spbsoftware.com), and Scramble. Another good one is Asphalt4, a driving game.
A couple other odds and ends: Personalizer (lets you change the Verizon Wireless banner to whatever you'd like), RotationService (gives you control over what apps can use the accelerometer - useful but takes some manual configuring) and Nitrogen media player (very fast, lightweight and good looking music player). WMWiFiRouter takes your Omnia's internet service and turns it into a WiFi hotspot for a laptop, iPod touch, etc.
As for apps not used directly on the phone, My Mobiler lets you control your phone from your computer and CeRegEditor lets you edit your phone's Registry from your computer (useful for making tweaks).
Just keep in mind that while it's possible to change just about anything you'd like on the phone, it's not always super simple and takes some time and patience. But believe me, it's worth it in the end.
That's all I've got for now, but I'll be back if I remember anything else. But like I said, browse through the forums and the internet to see if there's anything else that piques your interest. If you've got any questions, search for the answer first, and THEN if you can't find the answer, post a thread here. Someone will be glad to answer it, we're all friendly and willing to help.
ps - you can't change your username unless you get an administrator to do it for you. Definitely a weird one
pps - the line in my signature is the custom ROM i'm using right now. You can find it and all the others in the Omnia forum at the top - Omnia upgrades. But only perform these upgrades after running the official Verizon/Samsung "CF03" update
here! Also don't touch anything that says CC20 or else you'll likely brick your phone. Don't be afraid of flashing ROMs though, I've done it about 20-30 times myself and it's not dangerous at all.