Flavorings, Nic, Pg, and VG, obviously. Choose your nic based on what you usually vape. If you vape high PG, get PG based, etc. I usually do 70/30 PG/VG so I get my base in PG.
Something to measure. Syringes and needles are the must have IMO. Preferably in various sizes. 3 and 5 ml are good. I wouldn't mind a 10 ml for large batches, but I can live without it. I really want a 1 ml for small measures, but the 3 ml has 0.1 increments, so it works well too. You can add other stuff too, but you don't have to. I have pipettes and I like them, but most people don't. I don't see a need for graduated cylinders, but some like them. Droppers have their place for some things too, but again, are not necessary. Grab some syringes, and you can add other things to play with and test out in the future.
Bottles. A variety of sizes are good, but a limited range can be good too. I have a few 30 ml bottles from purchased juice, but I usually limit 30 ml batches to tried and true all day vapes. 10-15 ml are my favorite size. 6 ml are OK, but I usually use these just to pour from a larger bottle to carry with me, and not very often. When I go to work my whole collection usually goes with me. I consider 3 ml bottles worthless, since I can vape 3 mls in a few hours. When I do small test batches I just use a larger bottle. Then when I get the recipe right I can just finish filling the same bottle.
That is all that I consider mandatory. As your collection grows you will decide what your favorites are and what you can't live without.
ETA: Forgot the vendor recs. I have limited experience, but:
My first order was from onestopdiyshop. I got one of the kits cause even though I didn't think I would need everything in the kit, I thought it was easier to get started that way. I was happy with my order, but their selection is limited (like only small bottles of nic) so I chose not to order from them when I needed more.
Capellas- Great flavors, easy to use for those of us new to DIY. More expensive than other flavorings, but worth it for the better flavor, and still way cheaper than buying juice.
myfreedomsmokes.com- Good prices, good selection, quality products. This will be my go to supplier.