Welcome,
I had a lot of fun learning about the dark art of tech'ing a guitar - the basics are straightforward, but understanding the finer points takes practice.
I put together my own little toolkit - similar to this ludicrously overpriced fender one : http://www.takealot.com/fender-custom-shop-toolkit-electric-guitar/PLID42940749. Used a old pc toollkit for the case and basics and built up from there.
In there I've got :
Small shifting spanner
Small screwdriver
Multi purpose screwdriver + many bits
Wiresnips
Small pliers
Craft blade
Allen keys for trussrods & bridge saddles (that fit my guitars)
Stringwinder
a few miscellaneous bits - small pencil to mark things/graphite source for lubricating nuts, spare tuner battery, cable ties, earbuds, wire & solder.
And a leatherman, mainly for the pliers and super sharp blades.
Don't ask me what the drum key is doing in there...I guess you have to have something to throw at the drummer 8)
This is what I'd take to a gig. Probably with a soldering iron and a can of contact cleaner...only because I won't need it if I take it along.
This is a subset of what I've found useful when mucking about with guitars though...Also pictured is a drill w/string winder attachment my mate 3D printed - if you plan to work on guitars, something like this drill is a good idea - Imho, it's a must.
Truss rods are usually Allen key based, so a set of those would suffice for setting relief - though I'd rather use something like this : https://www.amazon.com/Fender-T-Shaped-Truss-Rod-Tool/dp/B003AYP9B8. Less likely to scar the area around trussrod hole when you adjust. Sure a local hardware store will have something similar.
Aaannndd...get a cheapie to practice on. A R500 Cort strat copy/SX/bullet strat, or whatever you find going.
The ES137 is valuable enough to have a pro work it? Maybe ask them to give you a tutorial as part of the setup - that would be cool! Occasionally a fret may lift so it might not just be a truss adjustment. Though loosing it a 1/4 turn may help (or a little more, but less is more with truss adjustments).
Happy teching!