Okay, now I'll try to delve into a "Must Have" list for striking gear. Remember, this is for a situation where you have yourself and a training partner. I'm going to address this from the standpoint that you are training a full 8-point offense system that encorporates Knees and Elbows, not just Punching. With that, let's see what I can come up with...
Head Gear (Simple boxing head gear will not keep you from getting KTFO, but will lessen the chance of superficial injuries like cuts and bruises. fat lips are one thing, but a bad gash on your eyebrows is lot more to deal with in training or afterwards)
Mouth Piece (Again a simple single-sided mouth piece goes a long way to avoid dental injury, and is a must for any contact/combat sport)
Supporter and Cup (If any sparring is done, this is a must for obvious reasons).
Boxing Gloves (I would go with your generic 16oz training gloves. Having these big ass gloves will do wonders for the actual speed of your punches)
Shin Guards (Simple cloth shin guards will help if you're trying to effectively learn how to throw MT style kicks, and also when you're learning how to defend such MT style leg kicks)
A Large Target Shield (With a training partner this can be used to improvise for a lot of other equipment: MT Pads, Focus Mitts, Heavy Bag, etc)
Heavy Bag (Optional if you have a readily available training partner and a Large Target Shield--Mandatory if you have a shortage of training partners or time)
Wrist Wraps (I've never gone into a situation where the punching was so intense while training that these were an absolute must, but if the focus of sparring is going to be in a 70/30 bias towards striking they would be recommended)
Instep/Foot Guards (If you're kicking properly with power, you'll be connecting with the base of your shin more than you would your foot, even then with the more effective thrusting kicks, you'll be connecting more with the bottom of your foot)
Ankle Support/Wraps (This will vary per user. If you have a history of bad ankles, I would say this is definitely recommended)
MT Pads (If you have a readily available and skilled training partner, these will do wonders for you. If you're strapped for cash, just get a large target shield or Focus Mitts-below)
Focus Mitts (See above. If you're trying to decide between Focus Mitts and MT Pads, go with Focus Mitts. You can do everything with Focus Mitts that you can do with MT Pads, and it will require that you be more precise when you train)
It doesn't matter if you agree or disagree
... As long as I don't bore you and I spark a moment of thought, my goal is achieved
Queng leon queng tigre ecu tacacut, queca pa? - Pampangan Mandarigma Motto