What type of wire for subpanel in attached garage?
The problem with NM-B and conduit is that you have to treat it as a single round wire the diameter of the largest dimension of the oval (or "round-ish" for 3 conductors and ground) for fill calculations - and, let's face it, it is a giant pain to actually pull. Also, it's a dry-location-only type wire and garages are skirting the edge of that being true.

I'd strongly suggest that you should just run conduit the whole way and then use THWN, XHHW, or the like. It will be better protected, and it may also be cheaper (large NM-B often has a significant premium over 3 or 4 conductors of equivalent conduit-type wiring - and the conduit you will use can be much smaller to run 3-4 AWG 6 conductors than one AWG 6 NM-B.)

4 THWN or 3 THWN and a bare ground (AWG 6) will fit in a 3/4" EMT conduit (per code-compatible fill calculations.) I'm having a bit of trouble tracking down a hard number for the diameter to use for NM-B in AWG 6, but I'm fairly sure it comes out to a much bigger conduit (based on checking the same thing in smaller wires.) You might want to run larger conduit for an easier pull, or stick with smaller conduit for cheaper tooling and conduit, or go with PVC conduit if you decide to go larger than 3/4" so you don't need tooling (at least in the form of benders) though you should be able to rent those pretty cheaply for a day if you wanted to do 1" or 1-1/4" EMT for an easier pull.

A $300 (or rent it for a day) drill option: If you already have a cordless drill (so you can skip the batteries and chargers) you can often get the right-angle drill to go with it for under $100 - might take a bit longer but will get your job done eventually if you'd prefer not to rent, without sinking professional electrician tool money into the big guy without having professional electrician income to offset it. Will also be smaller for an easier fit into your tight joist spaces.