The simplicity of a break open single shot action recommends itself to an absolute beginner. But since you obviously have some familiarity with firearms, it probably doesn't matter so much. Different action types have different pros and cons. It is important that you find what fits you well, as you probably already know from your handgun shooting, if you dislike a firearm for any reason, you probably won't shoot it very well.
The bolt action has become almost the default hunting type, so if "beginner's" means there is a significant chance you aren't going to keep it, this might be the type to choose with an eye to selling it on when you fancy something else. (But only if you are an unsentimental type - as a beginner, taking your first deer with it may create too strong an attachment to a rifle to allow selling it. I can't sell mine, which is also my first Lee Enfield and was a Christmas present from the then girl-friend.)
If "beginner's" means you don't want to spend a lot in case you decide not to like it that much, or want the scope to enjoy upgrading later, the low cost options new are the Stevens 200 bolt-action and NEF Handi-Rifle which are available in a good selection of calibres for well under $400. (And perhaps the Mossberg ATR100, but they come only in .270Win and 30-06, and don't seem to have attracted any great reviews like the Stevens 200, and they aren't much cheaper.)
For a little more you can have an H&R Ultra single shot, which is essentially a nicer NEF, or a Savage package of their model 11 or 111 rifle already fitted with an unremarkable, but probably satisfactory scope, and if you are careful you will probably find you get their desireable Accu-trigger with it. There are entry-level versions of the Remington 700, the discontinued ADL and its successor the SPS which aren't much more. Just be careful not to fall for Remington's answer to the Savage package, which I won't even mention the name of for fear of provoking one the loathe-fest threads that always seem to arise when beginners ask about the 710. Oh $%*!, I've said it.
Sorry.
Stick to commonly available calibers popular for hunting in your region. Even if you really fancy something exotic, that can be an excuse to have the fun of buying another gun later.
The other way to keep costs down is to buy used. What to look for in terms of type, make and model is the same as for new guns. If you are confident you can judge the condition of a used gun, and you are patient, you will probably find something suitable and bargainesque, perhaps even right here in our very own Exchange Forum.
And if money is no object, well, the choices just proliferate, and you probably don't need to ask then, do you?
Good luck, and good hunting.
