The documentation from Tesla's website specifies 6 AWG copper wire with 4 conductors (including ground, 2 lines and neutral). Since we have the standard range model vehicle, it can only draw a maximum of 32A from the wall.
Tesla's website is just an opinion. The controlling document is first NEC, and second the UL-approved instructions that come with the EVSE.
What matters is the ampacity on the (once again) UL approved nameplate on the charger, not some sales literature about what the car is supposed to draw. I suspect the nameplate actually says 39-40A. For any EVSE, this number must be derated 125%, so we arrive at 49-50A.
Your cable will include 2 hots and a neutral. It is a complete mystery to me why an EVSE needs neutral, but the upside of a 14-50 is if you ever get an RV, it can plug in there too.
NEC says a NEMA 14-50 recep can be on either a 40A or 50A breaker. The wire must be large enough for the breaker chosen.
When we bought the house there already appeared to be 120V electric run to the detached garage via existing conduit. We were hoping to dig a straight trench and run another conduit to house the new wire.
OK, stop and evaluate the existing conduit. It is certainly better to re-use the existing conduit rather than have to lay a second one. In fact, you need to know what's there now, because you are not allowed to have 2 same-voltage circuits to the same outbuilding. If the existing conduit is big enough, it may be better to install a subpanel and pull cable big enough to support the EVSE and also the other garage loads - typically 10A more than the EVSE will suffice.
Worst case let's say the existing conduit is 1/2". And let's say your EVSE's nameplate really is 32A - which must be provisioned to 40A. And let's say you fit a subpanel supporting a 40A EVSE breaker and two 15A breakers for garage circuits. You only need a 50A feeder for that (some oversubscription is allowed), and that's three #8 THHN wires and a bare #10 ground, and those just fit inside a 1/2" conduit.
Obviously if the existing conduit is bigger you have more size freedom. 3/4" = #6 THHN wires (good for 65A). 1" = #3 THHN (good to 100A).
Note that THHN is allowed to run at higher temperature than NM/UF cable (75C vs 60C), and so it's allowed higher ampacity (in NEC 310.15(B)(16)). That's how we're running 50A on #8.
With a bit of googling it seemed safe to dig a 18" trench and use 1" PVC conduit directly from the building to the garage.
You need 18" of cover. So for 1" PVC you'll need a 20" trench. 1" PVC is overkill, but overkill makes pulling easier.
I'm trying to find a good seller online since most local places are closed and both Home Depot and Lowes don't seem to show exactly what I think I need on their website (Unless I'm not using the right searching terms).
A real electrical supply house should be able to easily accommodate all your needs. Those are all over the place. You drive past 5 of them on your way to work. You don't see them because they're not in malls, they don't have 1000-space parking lots, they don't run ads on TV or have inserts in the Sunday paper. Their prices are better than big-box, and they'll work with you if you call them (most of them, anyway). They won't give you much help, but (huge upside) they won't give you BAD help like the staff at big-box who really know nothing.
Also electrical supply is cheaper for 90% of stuff, if they will give you trade prices. Some will give novices "list" prices; those guys are being jerks. I like to open the conversation with how I'm fed up with paying Home Depot's high prices :) (which is true and they know it).
Also Home Depot sells #8 and #6 THHN both by the spool and by the foot. You need a white neutral, but both hots can be the same color. If you're trying to cram #8 into 1/2" conduit the ground must be #10 and bare and copper, or it won't fit inside legal fill limits.
If you run #6 THHN you must use a #8 ground, for arcane reasons. #8 ground is good all the way to 100A (#3 copper).
Is there anything wrong with my thought process here? Something I am missing? Anything I can do to simplify, lower the cost, or make the setup safer?
Well, you have better instincts than most. I'm not worried about you doing anything unsafe. But still, "trying to fill knowledge holes using Google". Which doesn't work; Google doesn't tell you what you didn't know to ask. But you're here, so you'll get that if you keep reading and asking.
Harper's Rule: Buy the wire last. You'll learn more that will further educate you; don't pull the trigger too early and buy the wrong thing. Doesn't matter; the feeder goes in AFTER the conduit and panel work is complete and buried. (cannot assemble conduit around wires).
Cable inside conduit is a nightmare. NM cable isn't allowed outdoors (can't handle water) and UF is very flat, so it requires absolutely huge conduit. However THHN requires conduit the entire route panel to panel/socket.
As far as lowering the cost, the magic word is aluminum wire. It is your friend for large feeders like this. The only reason I discussed copper above is because of the desire to cram feeder into existing conduit, which is obviously a huge win if you don't have to trench it lol.
With aluminum, just use 2 sizes larger - #4 instead of #6 etc. If you use #4 or larger wire, it is no longer necessary for neutral to be white, it can be black like the others and mark it with white tape. Ditto ground (green tape).