How does tire size affect cornering? I've only had lower-mid grade 700x23c and 700x25c tires up until now. I would like to step up to some nicer(~$50-60+/tire) 700x28c if I can find some in my price range and fit them in my frame so that I can have a nice, smooth, comfy ride. Will the larger volume tires be more prone to feel squishy or wallow in the corners compared to the 23c or 25c tires when I'm going through sweeping bends at 25mph?

I am assuming clinchers from the sizes and price mentioned. The larger casing will flex more side to side if you accelerate out of the saddle. The 23's will all other things being equal, (often not) will corner with more authority, you will feel more in control up to the limit. They will also break free more abruptly. The bigger tire will feel more vague, (searching for a term) and may be as fast and corner as well ultimately, but you will have to adapt to the feeling it provides. I have found a narrower tire up forward with a high quality casing and a bigger less compliant tire in the back gets you both precise control and better wear in back where there is more load anyway. If you really want handling, go tubular.

If you can put the 23mm or wider tires on wider rims, and run at lower pressure, the overall feel is greatly improved.

I am willing to suffer the extra weight for being able to run the lower pressure, and the tire tread doesn't wear as fast of slip on gravel so easily.

There are a lot of new choices in wider rims, some are light. Even some of the older singlewall rims on 27" and 700c wheels only weigh a smidge above 400g though, and are strong enough for me. Others of this type like Mod58's are at least 500g though, but are still very fine rims.

I will come in slightly different on this one considering combarable rims if your looking for corning a fairly stiff or heavy set mid priced 25's is likely your best option. If your looking for overall ridning for transportation or recreation purposes on city streets and not so great bike paths buying a nicer set of 28's is a good investment.

I've actually got a set of Dura-Ace hubs laced to tubular Mavic rims. I bought them with the intent of swapping the freehubs from 6/7sp to 8/9sp and buying some nice tubular tires for the ultimate ride. Unfortunately more research showed that the 8/9sp freehub for this particular year of Dura-Ace is difficult to find.

This is my goal. I'm looking to have a nice, smooth, enjoyable ride on nice days when I feel like going for a decent ride and the MTB trails are too wet.

Tire callout dimensions are a joke. My Bianchi won't take real 28s, such as Specializeds, but Conti 28s, which are less than 26mm wide, work fine. I feel more secure on 25s than on, say, 23s, and I willingly accept a slightly slower, squishier ride if it reduces my chance of a catastrophic breakaway.

Your weight has a lot to do with it.

If you are too heavy for the width of the tire, you have to pump it up hard and lose some "gription".

I weigh ~150lbs. I've currently got 700x23c Vittoria Diamonte Pros at ~100psi(Minimum recommended pressure on the side of the tire.) My thought is that a larger volume tire will allow me to run a fairly soft tire pressure ~80-100psi while still giving me the tire volume to not pinch flat if I go over a large crack, etc, in the concrete. And hopefully absorb smaller bumps in the pavement as well making the ride smoother in general. I assume that's basically how the whole thing works with larger volume tires being a smoother ride?

It's been my (limited, so far) experience that tire pressure has more to do with the comfort/wallowing than the tire diameter.
Until very recently, I've always ridden 23's. And primarily Conti Gatorskins, because they lasted longer. Then I tried the GP4000's, which are squisher & stickier, and therefore more comfortable and are less prone to the unnerving scratching sound when you are banking hard into a hairpin on a steep/twisty descent. Seriously, I've passed dozens of cars on mountain descents and even four motorcycles, and I still haven't been able to get the GP4's to slip even a little. But that stickiness will dry out faster and crack more quickly than the harder compound of the Gatorskins.
I just recently tried 25s, and inflated them only up to 90psi--felt great on annoying, chip-seal type roads, but they feel dead in corners. In reading the above posts, "wallowing" seems the applicable word.
You'll have to find your own happy medium. For me so far; Conti GP4000s in the 23mm at 110-115psi seem to be it.
I still haven't tried the 25's at the usual 115-psi, but cornering at 90psi with them?--like trying to whistle the Blue Danube with a face full of novocain, or attempting to use the restroom while wearing boxing gloves.

I take it you've tried 'em at 90? That doesn't even come close to matching my experience. As long as the tires are of equal quality and similar design, I've found wider tires (at lower pressures) corner much more securely at any speed. Well, up to 40+ mph anyway. And this holds true for me from "700x23c" Contis (more like 20mm) at 120psi to 650x38b Soma B-lines (an honest 38mm) at 60 psi. BUT I am larger than the average cyclist at 6'3" and 185lb. I like to ride mountain passes and descend aggressively. As always, YMMV.

SP
OC, OR

I corner just as hard on my Bilenky tandem which has 42mm tires as I do on my Norman which is running 25mm tires. For me, tire size is less an issue than that of knowing my limits and those of the bike I'm riding.

BTW, back when I was riding motorcycles I had a 1993 CBR 1000F and I remember doing a two wheel drift as I was cornering with the bike leaned pretty hard doing around 60-70 MPH and with my wife on the back. I can't help but think that a lesser rider would have either tried to brake and stand the bike up and high-side or give it too much bar and low-side the bike. Either would have been catastrophic since a stone face where the road had been cut out of the mountain was what would have been slammed into with a wrong move. Admittedly, a low-side would have been preferable, but not by much as it would have ruined someone's day.