Beginner telephoto F-stop question

After picking up my first DSLR, the first lens I purchased was a Tamron 18-270mm. Even after picking up 3 more lenses, it's still the primary lens I use.

I don't claim to know much about photography (although I'm always learning a lot from these forums) but I thought this Tamron was a decent lens, until I saw this photo challenge...

http://www.dpreview.com/challenges/Challenge.aspx?ID=6632

200mm focal length at a minimum of F2.8?!?!?!!! I'm really gonna let my ignorance shine through here and say that I had no idea that was even possible. At 200mm, the widest aperture I can get out of this Tamron is 6.3!! Do you know how much grief I could've saved myself if I could shoot at 2.8? I do a lot of nature photography and I'm constantly having to adjust ISO and shutter speed to make up for the smaller aperture when I zoom, which is a pain when you only have a few seconds to capture the shot.

So, this realization has made me want to replace the lens I have with something better. However, as i said, I'm pretty ignorant when it comes to glass. What exactly am I looking for? If it has a larger minimum F-stop (say 1.8 or something), is that enough? How do I know how much the aperture is going to close as I zoom out?

Any help is much appreciated. Thank you.