I like AR's, I've owned one for about 15 years. I build AR's, sometimes in my sleep (I dream about it!). I've built a lot of them, including SOPMOD upgrades for my SDM students when the parts were available. I was taught by an SF gunsmith everything I didn't learn from Colt's manual. I have all the proper tools and gear.
Now, the last rifle I built was an M4 that is simply a close copy to the one I had in the army, but with a different trigger and suppressor mount. The one before that was a Grendel. A true tackdriver. The Grendel is the one I'd like to discuss as it is pretty close to what you want, but perhaps overkill. It is a 1000m rifle.
So, the most important parts are 1. the barrel, 2. the trigger, 3. the rest of it: good parts that are in spec or the best you can get, but they need to be reliable. Milspec at the very least if you are trying to do it on a budget. If it isn't milspec, or exceeds these specifications, don't bother with it.
For the Grendel, I used a stainless Satern barrel. They are very expensive though, this one was $400 and the .300BLK I just ordered is $550. Nothing compares though, nothing. Next is Krieger and Wilson. They make good barrels too, some folks may even say they are better than Satern (but I doubt it, he is one guy that does AWESOME work). The stainless barrel with cut rifling will give you the best accuracy, button rifling will suffice, but you want stainless for accuracy. Hammer forged and chrome lined, those are combat barrels. They last longer than stainless, but aren't as accurate to begin with. Stick to stainless. If you get a fluted one, make sure they flute the barrel stock before cutting the bore, and make sure they line the gas port up in a groove and not on a land, or worse, between a groove and land. Good luck finding a shop that will accomdate though, only one I know of for certain is Satern. Krieger or Wilson may do this though. And there may be others I don't know of. Satern is a small shop in Oklahoma or some such. DO NOT get a barrel by a cheap maker. Good barrels are only a little more than junk barrels, and some just don't perform that well to begin with. A $500 Sabre 6.5G barrel, tested against four other rifles, had the worst group using factory ammo. Worse than the J&T manufacturing rifle! The most expensive barrels generally are only slightly better than the really good ones. Keep that in mind, and go with a barrel maker that is known for good quality. I've named three really good ones for you.
The trigger that went into the Grendel was an RRA NM 2stage that came out of another rifle. I was going to put a high speed Geissele in it, but after playing with most of the high end triggers out there (Rainier Arms here has them on demo) I determined that my tuned RRA was as good or better than most others out there --without true measured lock times, you can't make an informed decision past how it feels to you, and that thread didn't go anywhere on here because nobody has the lock times published (or so few do that it doesn't help). Overtravel, creep, etc., that is what to look for. The RRA 2stage is also the cheapest 2stage trigger you can get and it is VERY durable. Works great. When I got it, it was a little "stiff". But after polishing the engagement surfaces (NOT removing material!) it is a very fine 3lbs or so on the 2nd stage, and super light on the 1st. The Geissele 3gun trigger on the other hand, the one in my M4, is super fast. Almost machinegun fast. Feels like a short one stage, although it is a 2 stage. Geissele makes great triggers in a great variety, if you know what you are looking for in a trigger, they are the ones to go to. If not, or if the selection isn't that dire, then just go with the RRA 2stage. I'm sure you'll be quite pleased. I wouldn't trade the one I have for a new Geissele, that is how much I like it. Once they are tuned, they are phenomenal.
You have receivers and small parts and such so you are good to go. You will need to find a shop to install your barrel, you can do the trigger yourself if you watch some youtube videos and/or get the Colt armorer's manual (surplus stores). Look around, prices vary CONSIDERABLY from sane to loco regarding barrel installation.
Your sights. I don't know what you mean by spring loaded... You mean factory sights? Your factory irons will work fine. If you want better irons, you can get a thin front post and NM rear. If you have a flat top, you can get a scope (if you don't have a flat top you can get a scope, it just works better this way). Don't get high magnification (most common problem, 4x or 6x work great on an AR). Don't get a cheap scope. Expect to spend about the same on the rifle as the scope for most rifles up to $1000, maybe a little more. A cheap scope can effectively neutralize all that you gained with the barrel and trigger.
Since you have a carbine, I'd get a full stock if you plan on doing long range stuff with it. They do make some, Magpul I think, that work like a full stock but use the collapsible buffer tube, so you won't have to get that. But when I use full stocks, I like the plain military ones. I prefer the A1 stocks to the A2 as they are a little shorter.
Now, since you have a carbine and you are getting a long barrel, you need handguards. You would be best served by free floating, that can really do a lot for long range work. Trust me big time on this one. I like Knight's Armament, the M5 FF rail, I have a few of them, but I also have an el-cheapo Oly FIRSH rail. Most would call it junk, I'd call it cheap and out of spec, but I have to admit it accepts the bipod just fine, it does free float the barrel, is just as accurate as the KAC given the barrel, stays on fine with loctite, and only cost $99. Oh, and I've had this one since '03 and I even used it in the army for bit.
If you want to convert that rifle to a varmint rifle, a true varmint style rifle, then you will be better off keeping that one the way it is and just building or buying a varmint rifle. All the big makers offer one. If you go to Rainier Arms, I can promise you they won't sell you junk because they don't carry junk. They'll also build a rifle to your specs, and if something won't work, they'll bring it up. Great guys, if I couldn't build my own, I'd certainly use them.
Another option is to order a complete upper and use your lower as it is for now. Later on, you can order a lower built to your specs, or just get one already built. Rainier can do both. Then you would have two rifles, price spread out over time, and wouldn't have to damage the rifle you have or end up with a bunch of parts you can't use. See, by the time you convert this into a true varmint rifle, you will have spend almost enough to do what I'm talking about.
Now, what kind of rifle did you get? Pictures? See, if you have a nice rifle like a Noveske or LMT or Colt, it would be a shame to modify it. If it is a plinker model though, or a lesser manufacturer, modify away! But if you plan on a lot of mods, it is better to just build it into another rifle over time. Been there and done that too.
Oh, suppressors. Yes, you can use a suppressor on an AR. I just put an AAC mount on four different rifles to use an SDN6 suppressor. I'll get the M42000 later on down the road. Now when you use a suppressor on a 5.56mm, don't expect quiet. Expect .22lr with no suppressor. Sounds a lot like 10/22 with no suppressor. The 7.62 suppressors will work on there too, they are a tad louder but the normal human ear can't discern it. The biggest difference is that the 7.62 suppressors tend to be deeper in sound than the 5.56. If you want to use it on more than one rifle, calibre, get the 7.62. Take a look at AAC, there are others, but they have a good mount selection and you can use the cans on lots of hosts. Now since the round still sounds like a .22, you may still want earpro. What the suppressor does do, is that it kills flash period. No flash. It also masks the direction of the shooter. It is quite difficult in the field to tell where an unkown suppressed shot came from. I don't know how useful this is in varmint hunting, I don't do it, but when hunting people it works quite well (I don't do that either, anymore!).
If you are interested in suppressed AR's, and decide that converting the one you have is not economical, take a look at the .300BLK. Awesome. Can't wait for my Satern bbl. This round can fire supersonic rounds at near AK ballistics, but fires heavy .308 bullets at subsonic speeds through a suppressor and (they claim) is quieter than an MP5SD! Good to 300m.
I have 'em in 5.56mm, 6.5G, .50Beo, and now .300BLK (soon). Plan on a 6mmAR and some other wildcats maybe in the future before moving on to the SCAR17 platform to see what I can mess up there.
I have a wealth of knowledge on AR's and such, if you have specific questions or need, PM me and I'll try and answer it, and if I can't, I'll say so or point you to someone who can.
Sorry so long, just trying to help and I have more to say than this, so pick my brain while you can. Provide me with specifics and I can give you better answers.