I am used to a thicker solder (maybe 1.5-2mm) from courses in school, but everywhere seems to recommend thinner for tighter areas. However, I am starting with just loose wires and terminals on components, so is thickness that vital? Would the thicker solder still cause issues?
I agree with @KyleB, some 1.5 mm diameter totally works, and it's what I used for years, though I now prefer much much thinner solder (0.5mm diameter). But it really makes little difference to the joint; all that matters is that you're able to supply a appropriate amount of solder. That's more a question of manual precision than diameter, though lower diameter can make things easier, but some find thinner solder harder to handle, so start somewhere. If you feel like it would be easier to get a different diameter because you're regularly running into the problem of having too much solder on your joint or having to feed an annoying length of solder into each joint, then go for it.
The act of soldering is a craft. Even though I've been soldering for about 28 years now, I'll never be as good as the Electronics Technicians I know (that's a proper occupation, actually a family thereof, with a formal 3-years education here in Germany); so I just pick what works best for me, and so should you: start reasonable (not the cheapest soldering iron, electronics solder somewhere between 0.5 and 2mm in diameter), and buy wire as you need it.
On Digikey (mainly used right now to get an idea, to figure out model numbers and such,) they have Kester Solder brand wire, two of which are 24-6337-8814 and 24-6337-7614. The first is part of the 245 series, while the second is part of the 275 series. Would either of these be sufficient for what I have planned?
Buying USD 42 worth of solder seems like proper overkill at this point. Unless you're starting to solder jumper cables, a single solder joint will need less than 0.1 g of solder, once you've got the hang of it¹. I'd rather buy a smaller amount of less industrial-quality solder – so you have a realistic chance to last longer personally than your roll, and you get to try a different diameter, or chemical composition, if you want, later on.
Anyways, yes, these two are high-quality electronics solders, and one of them is even lead-free. Although lead-free solder has the reputation to be harder to use manually, I don't actually think that'll be very relevant to your use case. Both have exactly the same melting point, and that's a low one, too, so they are both equally easy to get to flow.
Health-wise, the lead-free option is certainly nicer, but, honestly, the trick is not breathing in eather solder's fumes, no matter how little or much lead is in there.
Is any cleaning required for my chosen application, namely wire to wire and wire to terminal on motors and sensors and such?
No. It might look nicer.
In the future, if I DO expand into PCBs and maybe even surface mount, would the 8814 model number above, as an example, still work fine without needing to clean the board?
I am trying to minimize the hassle of this, so I am trying to find a solution which is most streamlined.
Then don't overthink it now. You're not going to solder SMD tomorrow, and the moment you do, you will be ordering these components anywhere that also has more solder – when you need it. No need to stockpile.
¹ I'm trying to guesstimate what I use per through-hole pin on a PCB, by my past consumption, right now, and right now I think I'm at about 0.0094 g, so that your 450g for 40 $ would last me about 47,800 solder joints. That is more than I plan to do in a lifetime; I don't solder for a living.