Road Bike Tires - Newbie Question

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I got a OLD rode bike looking to make it work in the future. It's a old FUJI road bike I picked up from 10 bucks. I replaced an intertube and it runs decent considering the chain was bent and I have to bend it back but probably needs to be replaced.

My main concern is speed. I just don't feel like I'm getting the amount of speed for the effort that I put into my tires. I'm wanting to get some new ones but need some recomendations. The ones I have on there now are pretty old school.

They look like this:
http://www.bikesomewhere.com/images/big/veloflex_criteriumTubular.jpg

and I see a lot of new ones that I think may be better. I need to know if they should fit on my old frame or not. I'd appreciate any recommendations!