Place the subpanel wherever convenient for now/future use
I gather you're contemplating a subpanel right next to the main panel because the latter is full. That's fine.
However if another location is also convenient, feel free to put it there, complying with law of course, and with a preference toward keeping it out of the weather. (even outdoor rated panels do poorly outside, and that is far more so with GFCI/AFCI breakers, which are required nowadays).
Do take consideration for difficulty of routing the feeder from the main to the subpanel. I strongly recommend metal conduit - you may be more familiar with PVC piping, but EMT conduit is quite easy to work with once you get the hang of it, provides better physical protection, and provides a grounding path.
If you put the subpanel right next to the main panel, definitely use EMT, and put at least 3 pipes between the two panels. There are advantages to having these pipes be <24" long.
Get a B.I.G. panel
I mean in terms of spaces. If you lurk around here, you've certainly heard our standard advice. And since I gather you're out of spaces, you certainly understand the problem! Right now, you can have a couple dozen spare spaces for the price of a couple of pizzas... and zero extra work. But later... that'll cost you the price of a panel and a day + of solid, regretful work. Yowch.
My "go-to" subpanel is a 30-space. They can be had very inexpensively.
Of course that means a panel with a main breaker much larger than 60A. That is fine. There is nothing wrong with feeding a 200A panel from a 60A feed breaker. The 60A breaker will assure the 200A panel does not overload. And since you're in the same building, your panel doesn't even need a main breaker; a main-lug panel will be fine.
Make up your mind: cable -- OR -- conduit + wire
One trope we see a lot is people specifying conduit, and then wanting to run cable inside it. They go to cable because that's all they know.
But cable is very stiff, and is rather hard to pull through conduit. (and I bet that was your first thought when you considered it!) So it requires quite large conduit, and novices often get what physically fits there in the big-box store. This makes the pull even harder than normal, enough to make you swear off conduit forever! What a shame, it's wonderful when used as intended.
Cable also has two problems: Physical protection (cable can't provide this directly), and wet locations (standard Romex aka NM cable is NOT wet-rated). Conduit solves these - and EMT is better for physical protection since it bends rather than shatters.
If you lay conduit, which I recommend - then you are better off putting individual wires in the conduit. These are sold in spools in individual colors. The wire is colloquially called "THHN", but the wire actually sold is dual-rated THHN/THWN-2 (dry and wet). Sometimes you see XHHW, which works too.
For size 6 AWG and below, you need colored wire - black for hot conductors (and using 2 blacks is absolutely fine), white for neutral, and if you run in PVC conduit, green for ground. (EMT is the ground). For size 4AWG and above, you can buy all black and mark the wires with tape on both ends.
You can use copper or aluminum for the feeder to the subpanel. If you use aluminum, you must "bump" 2 numeric sizes (remember smaller==larger). Aluminum got a bad rap because of inappropriate use in small branch circuits, but that never applied to feeder. It's a good choice for feeder. Price it both ways.
If the subpanel will be right next to the main panel, I would use copper, just because it's easier to bend (smaller gauge) and while significantly more expensive, you won't use enough of it to care.
You seem concerned about voltage drop. We don't even need to think about voltage drop until the wire length exceeds 120' for a 240V/feeder run, or 60' for a 120V run.
But even so, calculate voltage drop based on the actual normal loads, never, ever, ever on the breaker rating. And forget 3% - it's a good rule of thumb but 4-5% is also acceptable if there isn't too much other voltage drop. Especially, the normal wire will often calculate out to something like 3.4%, and the online calcs will recommend a size bump because of 0.4%. That's silly.
Feeder wire size
If the breaker is 60A, you need #6 Cu or #4 Al. Why? Looking at table 310.15(B)(16), we are required to use the 60 degrees C column, because the circuit is <100A. (>=100A allows 75C). Both these wires rate at 55A, and we are allowed to "round up" to the next available breaker.
- For a 70A breaker, #4 Cu.
- For 80A, #2 Al.
- For 100A, either #3 Cu or #1 Al. (It may be weird to get 30 extra amps for a 1-unit size bump, but remember - we can use the 75C column at 100A).
2" conduit is a very good choice of size, and will make for an easy pull in any of the wire sizes we discuss here.
And for a novice, first-time pull, using #6 Cu is fine. It's easy to work with, it'll land just fine on a 60A breaker without pigtailing, and the project will go smoothly. However, you do need to buy natively colored white wire for neutral and (if in PVC) green or bare wire for ground. The ground can be undersized.
Branch circuit wire size
Remember we calculate voltage drop based on actual load, not breaker rating. You are always allowed to upsize wire size.
With 120V circuits, my rule-of-thumb is a size bump every 60', but with an extra 30-40' thrown in because the circuit should never be loaded up to breaker rating. Below about 100', I don't worry about it.
The breaker size defines the circuit size.
14 AWG is only legal for 15A circuits, but myself, I do not use 14AWG anywhere. I don't even own any #14.
12 AWG is legal for 15A or 20A circuits. It is the minimum for 20A.
10 AWG is the minimum size for a 30A circuit, but may also be used for 15/20A circuits.