my most used tools with small engines(small engine mechanic professionaly for a few years)
get a good flashlight, i love my stylus pro, carry it everywhere. made by streamlight. i have several variations.
get a good rechargable work light, batteries get old after awhile. bayco has nice ones and are affordable.
a good spark tester, i get them from ebay briggs brand. you will need a special kohler one but should check with employer on getting one for you.
1/4" socket set standard and metric (mostly 10mm)
3/8" socket set standard and metric(most metric is 10mm or 18mm(1/2" drive) for blades)
1/2 " sockets but a set of short impacts, deeps not as needed, chrome is ok, but not needed as much.
standard set of wrenches up to 1" are good
metric wrenches from 10mm to 19mm are usually more then plenty for most use. 10mm being most common.
couple of crestent wrenches
fuel line clamp pliers, i get mine from ebay i think they were cal van tools. need those
i forget what the large size of flywheel nuts are, but those are good to have for impact use, also these should bee deeps as sometimes its handy, but shorts are generally used as most of the newer stuff doesn't require deep wells.
electric or air impact 1/2" drive
1/4" impact or electric for cover screws and the like, removal only, speeds you up.
most commonly found in my hand though is a 1/4" snap on racheting driver the one that accepts 1/4" sockets. love that thing, makes lilfe much nicer. get one!
get good screwdrivers, also get torx drivers i reccomend sk or any high qaulity brand. the sk have a good handle and are fair priced.
torx/allen/flat head/phillips socket bits. get all of them, they are very very helpful. i use the phillips on tecumseh carbs often.
good pliers, don't mess with wore out pairs. get nice ones, i reccomend channelocks.
vice grips are nice but not always needed.
beleive it or not, a pipe wrench, i use it to straighten a lot of things rather then for wrenching with.
hammers, get two big ones, they will be used to straighted decks. get a medium and small one for other lighter work.
good punches and chisels, i have craftsman they work. i use punches more then chisels.
for starters on alot of engines, swivel sockets are helpful, standard most of the time, but some have metrics holding them on.
get a long grabber tool, the one with the 4 fingers, also know as a glommer used for glomming on to things.
magnetic pick up tool, used all the time.
long long screw driver, get it cheap used for cleaning gas tanks and running fuel lines on tecumseh engines.
small torch, used now and again.
good pullers are good to have, but depending on shop they may provide them. so hold off till you know what you need.
get a grassbuster scraper blade, nothing but a grassbuster, it kicks the crud out of any putty knifel. made especially for mower decks. worth every damn penny. google it and you can find it in several places.
also nice is a blade buster/breaker. its a clamp for mower blades, handy if you don't have a impact for blade work. save your fingers, don't use blocks and don't hold it with your hand. i learned that lesson years ago. please take my advice be smart.
i think that covers my most used tools and some of the specialty stuff.
don't worry about things like leakage testers,compression testers or carb testers, shop should have those on hand but if not look around for good stuff.
listen to others that are in the shop, learn don't think you know it already. keep your head on straight and you will do fine.