My first was a Marlin 60 with scope, that was about 30yrs ago & it still shoots fine.
It is now my daughter's first gun.
Just a sugestion, if you are not wanting to go the .22 route there is the new .17 cal rimfire guns (.17 HMR, .17 Mach 2 & HM2).
Most are super accurate out to 100 to 150 yds they would be great for hunting small prey or just some super accurate plinking.
If you are the point of needing to save to buy a first rifle a 22LR is probably your best option. They are inexpensive to shoot, so you can shoot it a lot, which is most of the battle when it comes to rifles.
Note, I'm going by the fact that you said "first" rifle. If you are already familiar with rifles and rifle marksmanship, then ignore this post.
For a first rifle, I would suggest either a Ruger 10/22 or a Marlin 795. The Marlin 60 is good too, but the tube fed magazine can be cumbersome at times. It is cheaper, however.
There are Tech sights available for all of those (the 795 and 60 can use the same model Tech sights). These are basically sights that give your .22LR rifle the same sight picture as the AR, and many other rifles.
Then get in on an Appleseed shoot. You will learn tons about rifle marksmanship, even if you already have some experience.
After that, and after firing a few more thousand rounds, you'll have probably have a pretty good idea of what you're really after in a centerfire rifle.
i say .17 hmr is a great first. so much more versital. you can hunt small game up close, or a good distance out, such as cyote, target shoot. i shoot mine accurately at 300 yrds.
I'd also recommend the Ruger 10/22. They are a good inexpensive rifle that can be a great platform for learning to shoot. I have the tech sights on mine and they make a big difference and give the rifle the feel of a mini-M1A. My son has a Marlin 795 with the Tech Sights and it works very well for him, but aftermarket parts are not as common for those as they are for the 10/22.