I'm stoked you are looking at bikes. I hope my post is helpful and doesn't come across as condescending, I don't mean it to be. I've re-read it a few times, and I don't know how else to say it.
Those bikes you linked are pretty awful (no offense). The specialized could be ok, but I'd need to see it in person. Not all full suspension bikes are going to be better than a hardtail. I wish I had two or three pointers to help you know which full suspension designs are a good deal and which ones aren't, but there are just too many variables. You could find a great specialized for a great price, only to get it home and realize the rear shock is blown and you'll need to put $300 into an old outdated shock. It is possible to snag an ok bike at $500, but you've really got to know your stuff to know what to spot. I don't know of a great way to get you up to speed on all the old models, geometries, and components quickly.
It's kind of like the guy who wants to buy a wrangler for under $12k. He will come home with a YJ with a 3" body lift, 36" swampers, and stock gearing for $12k because it was $5k cheaper than a stock JK. To him, they're both wranglers. He thinks he got a great deal, but the rest of us know that YJ isn't worth $12k, even if it's in his budget, and even if it's a "wrangler."
True story. My sister just bought her son his first real mountain bike. She's stoked because it's a full suspension and 21 speeds and she only paid $75. It's an MGX. Little does she know that this "bike" sold for $100 new at walmart 10 years ago.
You're asking the right questions though. Bikes have been undergoing huge changes over the past 3 or 4 years. It's really tough to recoup much money these days selling a bike that's over 3 years old. If you're planning on spending more than 10 hours on the bike this summer, I recommend bumping up your budget and getting a great hardtail, or bumping up your budget a lot and getting a full suspension with the following features (in order of importance).
1) dropper post (this will put you out of the $500 range). This is a huge deal, because you lower your seat when you descend, and raise it when you climb. I use my dropper at least 20 times every mile. It's up, then down, then up, then down. It makes riding SO much fun. That being said, a dropper won't be possible for under $500. Droppers are pretty new, and they only come on higher-end bikes, so finding a used bike with a dropper will cost closer to $700-1000).
2) a decent fork from Fox or Rockshox. This will be easy to find in your budget, but I'd try to find one that was recently rebuilt. You don't want to end up with a fork that you can't get parts for anymore, or that needs to be replaced. Even finding a 26" fork these days is tough to do, let alone for less than $300 new.
3) the option for tubeless tires would be SUPER nice, but probably not going to happen in your price range.
It's tough to compare new bikes to old bikes; they've changed a LOT! You'll want at least 120mm travel. More travel isn't necessarily better than less.
In general, mountain bikes are categorized like this:
XC (Cross country). These bikes are super light. They're easy to pedal long distances. They rock on the climbs. They're great for fitness. They're not great for jumping, railing corners, hitting drops, or steep downhills. They're great for what they do, but i'm betting you'll want something a little more capable on the downs. In fact, I KNOW you will. These typically have travel from 80-120mm
Trail. (including all-mountain). This is the broadest category and probably what you'll want. This is a great compromise between climbing and descending. They're not the lightest bikes on the market, but they don't suck to pedal uphill with. They're decent on the downs, but not as plush or aggressive as an enduro/dh bike. Travel for these is 120-140mm.
Enduro. Not to be confused with enduro moto racing, it's totally different. Enduro is a category of racing where you have to climb to the top of the mountain, but the downhills are the only timed part. These bikes are usually pretty burly and strong and can handle big drops, but still climb pretty darn well. They're not "fun" on the climbs, but they get the job done. Enduro bikes really only started getting good around 2010. These typically have 140-170mm travel. They're the "coolest" bikes to have, but they're often too much bike for most people.
Downhill/DH. You can't pedal these uphill. They have gears, but no climbing gears. These often have triple-clamp forks like a moto, and unless you have a season pass to the canyons or deer valley, you won't want one of these. And you wouldn't want one of these that cost less than $5k anyway.
I'm not sure if that helps you or not, but I hope it does. Mountain biking is an expensive sport. VERY expensive, especially if you ride with guys with modern gear and you're going to compare to them. Kinda like riding an old 1982 xr250 vs a 2018 KTM300. Both will get you to the top, but it'll be a completely different sport/experience for those two riders.
Just for reference, if you were to buy brand new, the cheapest I'd expect a new rider to pay for a "good" plus hardtail with a dropper post is $1600, and a "decent" budget full suspension with dropper post is $2300. These bikes will provide a completely different experience from anything from 2008 or older.
If you just want to test it out and stay in shape in the off-season, you can find an awesome hardtail for $400, or a decent full suspension from 2006 for $650.
