It depends on the kit for me, but there are always a number of factors that are considered: Sound, quality of the shells and hardware, size of the bass drum and toms, how easy the bass drum tom mount is to take off and replace, condition of the shells and finish if second hand, as well as whether I like or can live with the finish. All of these, other than the finish, are show stoppers. It has to sound great, have good hardware and be easy to transport (see my comment on the Ludwig Club Date SE bass drum tom mount). Price is a consideration in that my budget is set before I start looking for a new drum set.
The first kits that I bought when I returned to acoustic drumming were bought with a limited budget so I had to make compromises. The Ludwig Accent CS Custom sounded great but the bass drum was not my preferred size...too long 22"X18". The Pearl Midtown was a nice compact size but didn't sound as good as the CS Custom. The Ludwig Club Date SE (Downbeat sizes) was outstanding but I found the tom mount a beast when gigging because it had to be removed to put the bass drum in a bag. That meant taking the mount off the rail and not losing the washer. Time consuming and I needed to carry extra washers. Those kits are gone and my budget allowed me to buy three outstanding kits:
Yamaha Live Custom: 22"X14" (Bass Drum), 20"X12" (custom cut Bass Drum), 10" X 7" (Tom), 12" X 8" (Tom), 14" X 13 (Floor Tom) - second hand no compromises, even love the finish. Only minor negative is it is oak and very heavy to carry.
Gretsch Brooklyn Micro: 16"X12" (Bass Drum), 10"X7", 13"x12" (Floor tom), 13"X4.5" (Snare Drum) - New with no compromises - wanted a more compact with a smaller bass drum. I like the finish, but the Silver Duco will yellow over time. Fits in two bags and is great for small venues and for jazz!
Gretsch Catalina (mixed kit with similar finishes): 18"X14" (Bass Drum - Mahogany), 12"X8" (Maple Tom), 14"X14" (Mahogany Floor tom) - second hand, bought because it was much cheaper in price than the others for outdoor gigs where it could get wet or indoor bars where I worry about the kit because the audience is dancing right in front of the bass drum. Kit sounds fantastic but I compromised on cheaper hardware and finish (mixed shell colors that are similar but not identical). I did not compromise on sizes, I wanted the 18"X14" bass drum.