GFCI is no-go with some transfer switches
There are some very terrible, very expensive transfer switches which have a row of switches, 6, 8 or 10 of them, and individual fuses or mini-breakers near each one. Those allow you to switch individual circuits from utility to gen (a "cute" feature but not worth paying $400 for, considering how junky the rest of the unit is). The 1970s called, and wants that design back. Those will not play at all with GFCI anything, and I can't believe it's legal to sell today.
However, another type of transfer switch looks exactly like a "subpanel". We typically advise you to build this out of parts, e.g. a 30-space Siemens/Murray main-lug panel e.g. SN3048L1125, their simple ECSBPK01 interlock (which is $25), and two common breakers. That will allow you to switch 26 circuits between utility and generator. All those circuits will be able to use GFCI/AFCI. If you want more circuits just use a bigger panel.
You mentioned a Reliance TRC1006CP9 in comments; that is the second kind and that will work with GFCI/AFCI. That particular model appears to be disappearing and I see some very good prices on close-outs.
Don't ever buy gear "for the electrician to use"
As far as the cable to buy, I get you are trying to "save money" by making the electrician use your stuff that you bought at Home Depot, which you believe is cheaper. However electricians do not like that, for very good reason: Usually, people buy the wrong stuff. SOOW cable is completely the wrong stuff for an in-wall installation, and if it was custom cut, into the trash it goes. Well, it can be turned into a nice generator extension cord.
SOOW is cordage which is flexible cords use for temporary connections. You know what cords are, you deal with cords every day. Cordage is only allowed for those applications, and cannot be attached or built permanently into a building.
In-building wiring is completely different, It is illegal to use it in the manner you would use cordage. It is only for attaching or building into a building. Which particular type depends on where your cable is going and how you plan to upgrade it in the future.
Don't confuse "gold-plated" with "copper".
One of the big mistakes here is to get Midas Fever and think that somehow copper is a better conductor than aluminum. Aluminum had some problems in a 1960s experiment putting them in small branch circuits... but for a big feeder in the 60A realm, it's the right stuff. Copper is just throwing money into a hole.
What's worse is when I see people spend lavishly on copper feeder, then chintz out on something that really matters, like panel size or use of conduit where it would help.
Cable has the advantage that it can run without conduit. It needs physical protection in certain areas where it'd be vulnerable to damage, e.g. the exterior of buildings, underground run stub-ups, or along garage walls.
Conduit is a good way to run wires. You use individual wires called THWN or XHHW, and use exactly the wires you need. #6 or smaller neutral wires must be white or gray. A ground wire is needed if the conduit is not metal.
However cable inside conduit is very awkward. That combination was never meant to work, and since the cable tends to be flat-shaped (especially UF), it takes absolutely huge conduit. #6/3 UF cable is 1.35" wide and requires 2" conduit (clearances would be too close for 1-1/2" conduit).
You can switch methods halfway down a run, but if there's a cable to wires (THWN/XHHW) transition, that must be in a junction box.
Conduit + THHN/XHHW wires allows higher ampacity from the same size of wire. For instance your goal of honest 60A requires:
- Conduit: #4 Al (rated for 65A) or #6 Cu (same)
- Cable: #2 Al (rated for 75A) #4 Cu (rated for 70A)
The disadvantage goes away at 100A. If your target was 100A you'd use #1 Al or #3 Cu in either cable or conduit. Given that your plan was #4 Cu anyway, it's silly not to take the "mini-bump" to #3 and get access to a full 100A. Except do it in aluminum with #1.
#3 Cu barely fits in 1" conduit. #1 Al barely fits in 1-1/4" conduit. If the idea of taking the mini-bump to 100A seems stupid to you, 1" conduit will do fine, or even 3/4" if you stay at #6 Cu.
Cheap: Cable the whole way, selecting cable appropriate for indoor/outdoorness.
Plastic-plated: 1" PVC Conduit the entire length and THHN or XHHW wires inside.
Silver-plated: 1-1/4" EMT conduit the entire length, going to PVC underground as EMT isn't allowed there.
Gold-plated: 1-1/4" Rigid conduit the entire length.