Having worked in my dad's TV shop for 30 years, here is the list of stuff I'd get first, arranged in most useful to least:
A Mirror - get one for the back of your bench and a portable one - handy for portables and the test jig. Use it enough, and folks just don't look right unless they are reflected in a mirror....
Ours were broken back of diplay case units from the glass place, and our portable was an unbreakable one about 16x20 or so used in prisons....
DMM - we were partial to the Fluke 70 series, but the beckman 310 could measure up to 1KV. I have the Fluke 77 and Beckman 4410
Cap ESR meter - if not servicing solid state stuff, move on. I have the Anatek/Bob Parker unit and a Sencore LC102
HV Probe - Get a Pomona (EDIT: model 2900) standalone with the integral analog meter. The optional fluke job is too bulky.
Scope - Anything beyond 20MHz is overkill for most TV repair. I have 3 - the Hitachi V212 daily driver, a Kikusui COS6100M and a Sencore SC61
CRT checker - B&K units made money for us - started out with the 440, and ended after 30 years with (4) 470s - one was a parts unit. We had all but one adapter - but used only 3 for most checks - delta/socket 3, and two different inline ones - one for the Asian stuff and one for the A63-series stuff. I've used the CR70 from Sencore at their sales seminars - it rejuv's just as good as any. Folks like the universal adapter, but B&K made one as well.
Ringer - I use my VA62 or LC102.
Convergence/Color Bar Generator - we used RCA stuff - WR504/ Wr-508. Skip the tube stuff. Leader also makes a good one - not sure of the model, but smallish and steady unit.
Degaussing coil - make your own with a momentary switch (rated at least 5A "inductive" - most 10A/125VAC switches will switc 5A inductive loads..). We had two - both were GC hard-plastic encased units, and we added 8' more cord, as outlets are sometimes scarce. Hobby Electronics magazines in the late 60s had plans - check with American Radio History.
Dedicated transistor tester - we started out with an octopus/Eico 460 setup, and ended up with a Cricket - handy for checking stuff quickly, albeit 90% of checks were PN junction checks with a Fluke.
All-in-one unit - We used them all - B&K 1076/1077 and both the VA48 and VA62. Handy for supplying missing bias, sweeping yokes and flys, and providing a sweep signal so we could "walk the IFs" with a scope. Most handy in supplying a known-good color signal to trace color problems, albeit about 90% solid state stuff (hint: always the jungle chip, crystal, or resonator...)
Substitution cap - take a big ol' 80uf/450V and put some clip leads on it - handy for finding open caps that cause 60 or 120Hz "crawl"
A good outdoor antenna - not everyone has cable, so you gotta check more than channel 3 or 4....
Alignments were RARE. I know of 2 we did in 30 years - one on a Sharp solid state set (open IF can, had to realign, used VA48) and a APC/ACC Color alignment on a GE tube set, IIRC. Easy, two coil alignment. A new crystal got us color back, but the phase was out slightly, yielding green hue at center of the tint control.
A good computer loaded with service info - for PDFs (we had 66 Gigs of manuals on PDF, mostly Hitachi, RCA, and LG/Zenith.), Parts Finder, Sony DAS, RCA Chipper Check, Zenith Z-tips (dos only...), and two stand-alone tip products we contributed to - 64,000 TV/VCR/Monitor tips at the beginning, not sure of after many updates...
And then the consumables - cleaners (we used Windex, Fantastic!, and Olde English wood cleaner)
Solvents - 91% IPA for flux and flushing switches, Acetone for degreasing, and Naptha for petroleum-based cleaning. In earlier times 1,1,1 Trichloroethane and a Rawn-chemicals HCFC product like 1,1,1,.
Lots of acid brushes, RTV sealant, Heat Sink compound (Dow-Corning, not the GC or Chemtronics stuff.)
Dad was a WD-40 FANATIC. We had the gallon can and the little spray cans.
Freez-mist from Rawn - a 20 spray Oz can would last a month or so.
I've probably missed alot, and others, feel free to add/admonish.....