If you don't mind input from a rude old man: I think the 30-30 would be an excellant choice for several reasons:
1) Tradition, everyone should have one. (Or two, or three, etc)
2) Enough rifle to perform your task of deer at 200 yards.
3) Ammo is plentiful and relatively cheap.
4) One of the easiest calibers to get a Handi to shoot accurate.(Along
with the 45-70)
5) Is a rimmed case, which some feel Handis prefer.
6) Once fired brass is everyware and cheap. (if handloading)
7) Reloading data and components are everywhere. (if handloading)
8)This caliber takes very well to cast loads. (if handloading)
On a further note: Although a lot of elk have put down with the 30-30, many would feel it is a little on the light side for this game. But, we all need shiny new toys to just enjoy. And I think the 30-30s (of all types) shine at this. I feel it is the perfect plinking caliber. More authority then the pistol rounds, but will not beat the stuffing out you on a long day at the range.
And, since this will be your first Handi: WARNING: If you are looking for a rifle that will alway shoot as desired right of the box, be prepared for a surprise. Some of them take some tinkering before some issues can be overcome. But, the 30-30, along with the 45-70, are probably the least likely to need that tinkering. Research some of the posts on issues that have shown up. There is lots of experience on this forum, and all will be willing to invest their time and experience to help if any issues do show up.
Just my thoughts. I'm sure others will chime in with their favorites. And, you can always add more calibers. By the way, welcome to the club of Handi fans.
Wally