Because you really don't know what you are doing, I'm starting to assume that you didn't setup your sequence correctly in Final Cut. When you first dropped footage in, did it throw red-bars on top of the editing window? Did you have to transcode the footage in order to see it?
If so, that is one thing that might be making your footage look bad. You are essentially creating a copy of a copy and then forcing to work in your edit. If you do that, your output is going to suck. You always want to avoid transcoding until you export the final... unless you do filters and effects.
If you are making master copies, then you don't need to transcode them. Just export a Quicktime using the sequence settings (if you set that up correctly!). Though others might disagree, I find that Quicktime conversion is not that good. It has a tendency to muck-up your video if you don't know exactly what you are doing.
When I finish a video, I usually export it as Quicktime (not conversion) and make the movie self-contained. If I do a lot of effects, or need to transcode the footage, I'll send it out as loss-less animation codec. The file is huge because it's basically uncompressed video. That master file I use for whatever delivery method I need. If I need to compress for the web, I'll usually use Apple Compressor to do the conversion. I even have a Compressor droplet on my desktop that allows me to easily drag and drop my movies for quick conversion.
Sometimes I send the footage to Compressor from within Final Cut and bypass the encoding in Final Cut. It all depends on how I'm going to use the footage.
Anyway, back to your issue. You are choosing to export with Quicktime conversion with H.624 codec. Normally a good DELIVERY choice, but it's optimizing it for web delivery instead of just making a master copy. If you want the best quality for viewing you need to use a different method such as Compressor to do the conversion.
I suggest you google Compressor and take a look at what it does. It seems awkward at first, but it works very well once you know the interface. Again, I rarely compress out of Final Cut unless I'm just saving master footage as defined in the sequence settings.
Here's a link to a compressor how-to video that shows you how to prepare a video for Vimeo. Keep in mind that this is going to compress your movie... it's not good for a master. You might find it informative to watch and it will show you the basics of using Compressor.
http://www.vimeo.com/2051588
Hope that helps
