Back to the basics?
Very easy to get very quickly and very deeply into the weeds here ;-)
Maybe there is value in looking a little bit at the basics? What are we looking for in that first gun? In some semblance of order, it needs to work; it needs to kill; you do not want it to hurt you; it needs to allow you to grow with it; you need to spend your money where it is needed; you need to fall in love with it, and maybe a few other things like it needs to fit. In a way it is like buying a first car or a first motorcycle; there are classic mistakes: buying too big; buying too small; buying too expensive; buying too cheap; not buying what you need; etc. Let's try to take a look at these...
Need to work. About most any modern gun design in commercial production will actually work, regardless of brands, whether new or used. Where things require a bit more attention is that modern mass manufacturing (i.e. low cost) and quality control (or lack thereof) will occasionally turn out a lemon, even in a revered brand or model. For example the last few years of manufacturing of the Winchester 70 in New Haven, on worn out machinery and with a demoralized workforce, produced some guns that were objectively defective. On one of mine for example, the safety simply could not be engaged. It only took me 15 minutes to fix it because I knew how to take apart the firing pin and I knew what edge to file, but to many people this would have been an non-fixable issue short of returning the gun. It can happen with any manufacturer. The advice here is to buy your gun, new or used, from a reputable dealer that can perform a warranty exchange or repair, should you be unlucky enough to have a problem. A first time gun buyer should probably abstain from purchasing from a private party, unless you know the person well, or unless someone can assist you in making sure the gun is functional before you pay. Observe that so far brands or calibers have not even been discussed.
Need to kill. Well, that means two things: accuracy and caliber adequacy. Again, most any modern gun will be plenty accurate for hunting purposes, but a few will not. One would say beware of used guns in hot-rod calibers that burn out barrels in a couple thousand shot, but you are probably not in THIS market, and most people will never shoot a couple hundred rounds in most hunting rifles. In a used gun, check the barrel. Shinny? Sharp edges in the rifling? No rust or pitting? etc. In a new gun, back to the dealer/manufacturer warranty discussion.
The second, and much more complex discussion here is caliber. Everyone has (a) favorite caliber(s). Try to leave that aside and analyze your needs. For example you say "plains game." That covers a myriad of animals from a 50 lb duiker to a 2,000 lb eland... Hmmm... Let us tie in the discussion now the 'you do not want it to hurt you' and the 'it needs to allow you to grow with it'. You see, most caliber recommendation you will hear are made based on the recoil generally associated to a cartridge. This is why you hear now .308, 7mm-08, 7x57, .270 etc. These are all mild recoil calibers and good suggestions. But this is not enough. A more complete way to look at it, is that recoil - which you generally want appropriately mild in a first rifle - is a direct function of caliber indeed, but even more: bullet weight and gun weight. For example, an 8 lb 7mm-08 Rem. shooting a 140 gr (grain) bullet at 2860 fps (feet per seconds) will produce 12.6 ft/lb (foot/pound) of recoil, while a 8 lb .308 Win. shooting 180 gr at 2600 fps will recoil 17.5 ft/lb. 39% more! You will feel it. But, the same 8 lb .308 Win. shooting 150 gr at 2600 fps will only recoil 14.2 ft/lb. You will likely not feel the 10% increase in recoil, but the 7mm-08 is essentially maxed out with 140 gr bullets that are too light for heavy game when things do not turn out perfect, while a .308 has a lot of room to grow if you start with 150 gr. The same concept applies with the more powerful calibers, to the point that a .300 Weatherby, for example, is literally two guns in one between 150 gr bullets and 200 gr bullets. The advice here is: do not buy too big, of course, but do buy smart/big enough that our gun can grow with you as you learn to control recoil and as you hunt different animals. And for crying out loud, if you are a reasonably healthy young lady, yon CAN carry a 9 lb scoped rifle, so stay away from the foolishness of a 6lb. gun that will quick you into the next county everytime you pull the trigger.
Spend your money where it is needed. Very simple here: you need to spend as much, and likely more money on the scope than on the gun. And spend money on the mounts too. You cannot hit what you cannot see. You cannot hit with the wandering zero. nough' said.
Get the gun you are in love with. You will not hit with a gun you do not like. You will kill like by magic with a gun that you trust. Yeah, really. Truly EVERYTHING ELSE (within reason, i.e. it is mere foolishness to shoot an elk with a .223) is secondary to that.
Oh yeah, we have not covered yet the 'it needs to fit you' thing. This is because, you see, countless kids of any gender have killed countless game with their Daddy's gun, that they drooled upon for 5 years, worshiped for another 5, and finally have been allowed to take on that first hunt after months of increasing hopeful anticipation. Never mind the gun did not fit them... Of course, by all means, do get a stock that fits you, but do not think that just because you are a young lady it needs to be pink or half the size of a man's gun stock. Fit has nothing to do with gender; it has to do with build, and many, many men are not built all that different from many, many women. Marketing geniuses not withstanding. Do not give up a "men's gun" there is essentially no such thing ;-)
Ah, but now we can get into the Savage vs. Remington vs. Winchester vs. blah blah blah LOL.
One parting thought: if you are not all that enthused about oiling your stock every year, cleaning your barrel after every shooting session, etc. give a good hard look at stainless steel guns and synthetic stocks. Not to mention that you WILL sooner or later hunt in the rain ;-)
Another parting thought: did you notice I spoke about used gun? Killer value (pun fully intended) in many cases, if someone helps you to buy, or if you can get a 10 day or a 2 day return-warranty (i.e. enough to go shoot a box of ammo with it). Spend half a day at the closest Cabelas or Sportsman Warehouse or such and touch a lot guns, new and used. You will know when you find the ONE.
Welcome!
While I'm freezing my balls off trying to get a rifle to shoot up here in the frozen northeast you're enjoying the sun. Thanks alot! 
