The M44 is a good choice as a first rifle for deer hunting, unless you're of really small stature and very sensitive to recoil, in which case a Chinese or Russian SKS might be a better option at a similar price (if you can find them). The 7.62x54R is somewhat similar to 30.06, versus the 7.62x39 fired by the SKS which is closer to 30.30 Winchester. A longer rifle like an M91/30 might also help with the recoil and provide a longer sight picture, but at the expense of the carbine's handiness in the bush. I picked up a laminate M44 from Tradeex last summer along with a K31, and was impressed with the quality of the rifle.
Currently, there seems to be a selection of soft point hunting ammo from Igman, Prvi Partizan and Sellier and Bellot (all from Eastern Europe), or even Norma if you're willing to pay for it. A few years ago Winchester was producing SP ammo in this calibre, but for some reason I haven't seen any myself. You'd think that the US ammo companies don't want people to buy these rifles over domestically produced ones, since I'm sure that there would be a market for SP ammo from those companies if they produced them. It might however simply be due to current restrictions on exporting US ammo in certain military calibres.
The 7.62x54R is a good round for deer and other big game, and you should easily enough be able to find a couple of boxes to sight in/familiarize yourself for hunting with it until you get into reloading. If you already practice shooting with a .22 rifle, a box or two of ammo will be enough to sight in the M44 and familiarize yourself to shooting it at closer ranges like within 100m. With its iron sights, someone with good eyesight and familiar with the rifle might even be able to kill deer at twice that range, but probably not without a fair bit more practice.
Aside from the folding bayonet looking funny, and the slightly clunky feel when operating the action when compared to a nice Mauser or Lee Enfield, an M44 is just fine for deer, especially in thick bush. One of Tradeex's M38s would also work fine (I got one a year ago) though I understand that their M44 rifles have mint bores whereas the M38s have clearly been used, even if still very good and servicable.
I really like the Laminate stock on my M44 and think it worth the extra $10 or so that it cost me. Anthony at Tradeex is a great person to deal with and I highly recommend his business.
Cheers,
Frank
Good Luck.