Well, the "SM57 of shotguns" is still the Sennheiser 416 on a nice lightweight fishpole with a furry hamster wind gag, probably a Rycote one. However, more a more location sound is now recorded in M-S stereo--I don't think there's a universal solution yet but the Shure VP-88 turns up on a lot of sets.
However, if you want to be fully equipped, you'll need several lav style radio mics that can be taped on and hidden under clothing and with battery powered receivers. My personal favourite are the ones by Audio Ltd.--the transmitters and receivers I bought in 1984 are still going strong with the guy I sold them to when I retired.
Mixers...do you really mean "on set" or are you looking for a more portable documentary style thing?
For real "on set" use (mainly dramas, something from the
AUDIO DEVELOPMENTS range is pretty much the industry workhorse. For big, fully-featured stuff, the AD149 is pretty commonly used--but it needs a source of mains power (not an issue on big sets with generators). However, just a bit smaller is the AD245 pico mixer which can run off internal batteries. Both have truly excellent pre amps and the sort of features you'll want on a set. I used the AD145 pico (predecessor to the 245) for years and it was built like a brick outhouse, never failed and sounded great.
If you mean the more documentary style with a mixer you can hang round your neck,
SQN is the common fully professional mixer...probably the larger stereo model.
Other things...for real "on set" stuff you'll likely want a trolley to carry the mixer, equipped with storage for mics, cables, etc. It's also now very common to be asked to provide a basic form of in ear monitoring for other members of the production team (producer, director, continuity, etc) to listen to dialogue on. You'll need a good set of headphones with excellent isolation so you hear what's being recorded, not spill from
the live sound.
One tip. Make sure you're sitting down when you see what all this costs. FYI, there are cheaper options but you asked about the "best".