Danauk makes a good point regarding mouthpieces.
Most trumpet players (and other brass musicians) should find one that suits the size and shape of their mouth and stick with it. While there are guidelines for different sounds such as high and bright or low and mellow, staying with one comfortable mouthpiece is best for all but the most serious players who can devote a lot of time playing with each mouthpiece.
I know too many trumpet players who constantly switch mouthpieces depending on the music or sound they want and miss a lot of notes because they simply don't play enough.
I've used a 7C since day one (about 45 years ago) and have never been limited by what many will refer to as a "student" mouthpiece, but I know exactly where every note is with that 7C and it has never failed me. If I need a brighter sound I switch horns from the Bach to the Yamaha and that works... for me.
I have some friends who also stay with the same mouthpieces (5C and 3C) and do very well. In fact, the one using the 3C exclusively is the "screech man" in the swing band I play in and uses his B&S for high and bright and his Bach Strad with the same mouthpiece for a broader sound. My philosophy is to develop the lip and stop switching mouthpieces.
Just one more plug for the Bach Strad: Just about every decent trumpet player I know has at least one Bach Strad.
Barb D: What work your old trumpet will need depends on how well lubed it was when it was last put away. You need to make sure the valves operate easily and that all the slides still move. Take it to your local music shop and they can usually get it going with a good cleaning and lube and they might even be able to nail down it's origin. I'm not familiar with "Frank" but that doesn't count for a lot. I have a couple of friends who have horns from the 1800's that are just as good as new horns.