First of all if this is indeed a cherry gun, I would hesitate to make any changes that can not be easily put back to original. There are a lot of these guns out there but not a lot of realy nice ones.
If you want to go to a centerfire, remember that the 44 action has limits. The .25-20 SS or .25-20 Winchester or .32-20 with light loads would be suitable. I would not go with any hotter centerfire.
I would consider rebarreling, either with a new barrel or an original barrel relined. I have one that was .22 and I had it relined for a .22 Magnum. I have another one that I bought already converted to .17 HRM. Both are appropriate for the strength of the action and fun to shoot. I had to have the firing pin slightly relocated going from .22 LR to .22 magnum. The .25 MAY work without modification.
If you go to a centerfire you will either switch to a centerfire breech block or have to firing pin moved to the centerfire position. If you rebarrel and get a different breech block, you can always switch back to the original .25 Rimfire if you wish. I think the gun would be worth more at resale if it can be swithced back to the original configeration. John Taylor, who is often on this forum, did my .22 Magnum conversion and is as good as they come and a good guy to deal with.
The trigers can be reworked by a good gunsmith. Got to be careful because I have one that someone got almost too light. Touch it and it goes off. It is one of the most accurate guns I own, largely because of the light triger pull, but you have to be very careful with it. It is close to being dangerous. Hope this has been helpful. I am sure you will get more input from others here.