First handgun must be ACCURATE, and have EXCELLENT sights and trigger. Period! There is nothing worse than learning how to shoot and getting hits all over the paper, just to discover that ammo/handgun combination alone shoots 6” on 25 yds. With addition of blurry sights and jerky trigger, shooting could be so frustrating that it might ruin any further desire to continue with this sport. Been there, done that! Big mistake on my side because I did not listen others who went through all this before. I went el-cheapo route, just to waste one year and enough money to start with appropriate handgun at the beginning.
In that respect, when purchasing first handgun, I would suggest to look for a good used one and make sure that you test it before making final move. Try as many different handguns as possible. As for testing, if you find handgun you like, get somebody who is a good shooter, preferably with same handgun. Something that shoots OK for one person might not be best for another one. As others suggested, look for a good 22 first, and stay away from single action unless you can afford Freedom Arms. Look for one of those that have all sights fixed on barrel/receiver. Ruger 22 is a very good one, see also Margolin
http://www.militarygunsupply.com/images/firearms/22mar.jpg . Later one could be on the market under other names but is a very good one and affordable. Also, some of earlier S&W 22 revolvers are also excellent shooters and not so expensive as later fancy models like 617. The advantage of revolver is usually excellent trigger, no take-up as on semiauto pistols. If you ask for my opinion, get as a first handgun 22 TC Contender. It goes for about $400, it has rifle’s accuracy and excellent trigger. Once you do not need it anymore, you can always sell it for the same money.