You can't go wrong with Glock, Springfield, Smith & Wesson's M&P series, CZ75, or a Ruger SR9. However, Sig has a few models that are around $600, and you can get an XDM for 600$. That extra $200 opens up a whole new world - you could probably find a used SIG 226 for around $600 on your regional Facebook firearms group. I'd probably stick to a Glock, Springfield XDM or S&W M&P and get at least a 4.5" barrel as a beginner, and only go Ruger or SIG if you find a good deal.
Do yourself a favor and go with 9mm. You'll get superior capacity, and 9mm gets the most attention from ammo manufacturers. Some of the new rounds for a 9mm create 440+ lbs of energy at the muzzle - rounds like the 147gr Ranger SXT that LE has long sworn by only have 320, and I trust those completely. I have been impressed with Liberty's Civil Defense, and Mag Tech's Justice, which are 55gr and 92gr respectively, but have muzzle velocities of approx. 1900fps and 1500fps respectively. I have tried ballistics gel, ribs wrapped around water jugs, and a few other testing media, and have gotten consistent results with everything.
If you want to go bigger, go straight to .45. .40 is an answer for a question that doesn't exist. They are harder to resale, the bullets are often harder to find (except when everyone makes a run on ammunition), and most of the original .40 caliber guns were built on 9mm platforms, meaning your firearm will not last as long, and is more prone to a catastrophic failure. Don't switch barrels either - it's alright to get a .22 barrel for plinking, but buying a .40 GLock just so you can switch between 9mm and .40 is not a good idea. You sacrifice accuracy, and it increases the likliehood of a failure to feed, eject or cycle. If you want to shoot 9mm, buy a 9mm.
Whatever you buy, don't be afraid to shoot it and dryfire, and get some type of comprehensive basic training that teaches safety, stance, grip, sight alignment, and trigger pull. I always replace the triggers too. All these guns can be dry fired (although the XDM has problems with the Striker retaining pin breaking if you dry fire without snap caps a lot - Powder River Precision sells replacement roll pins that solve this problem).
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As a side note, I put together a small list of learning resources for first time gun owners, and people that have owned firearms for years but never bothered to master the fundamentals. The first stage of learning a skill is unconscious incompetent - you think you're great, but really don't know wtf you're doing. Learn about the
4 stages of competence . This is an extremely pervasive problem with gun owners. I looked up some basic instructional pages, and picked out a few good resources that have sound, proven, and widely accepted instruction.
Fundamentals of Pistol Shooting - Gives basic info and advice
Fundamentals of Handguns (This is a Comprehensive Guide - Be sure to read this)
Travis Haley Teaches Proper Handgun Grip
[video=youtube;DDZDttBfock]https://www.youtube.com/watch?v=DDZDttBfock[/video]
If you are serious about becoming a good shooter and building on solid fundamentals, I strongly suggest purchasing better education materials.
Magpul Art of the Dynamic Handgun Video- Basic Marksmanship followed by effective drills
Donn Mann's Modern Day Gunslinger -This is considered the Bible when it comes to Pistol Marksmanship Books. Has good pictures, and gives a good account of the history of marksmanship development.
After you start practicing, this illustration sometimes helps diagnose problems. You can self diagnose problems by eliminating each variable one by one - this chart will not always give the correct answer
I'm not saying that OP is not a good shooter, or that anybody isn't, but if you haven't learned fundamentals from an authoritative source, you are not shooting as well and as efficiently as you can. The majority of gun owners think they are good shots, but have a severe lack of knowledge. Being honest with yourself is very important if you wish to improve your performance with a handgun. There is nothing wrong or embarrassing about not being an expert.