(Caution: More detailed response than necessary, in light of you asking the question John, and the hope that we'll still be able to get together soon and do a rec climb or two and/or a couple small work climbs, since we are not far from one another. I like your interest in climbing and would enjoy sharing the knowledge and skills I have developed to help you along your way).
Pruning: Saddle, foot and knee ascender, helmet with standard ear protectors (no radio - I prefer not to be chatty while pruning in the zone), 4SRT or Didier chestie with XSRE mini-carabiner as a HUT, lanyard, handsaw, blood-stopper pack. Depending on the tree and situation a second SRT device (or hitchcord/Hitchclimber pulley) to rig the tail of my line. In the hot part of the year I'll carry a water bottle on my harness. I have a small bag on the back of my harness. Depending on the day it might have my phone, a few energy bars in case I want to finish and not come down for lunch, a throwline reel, grapple, or a couple runner loops/carabiners for setting redirects, transfers, etc. Saw stays tied on the tail of the to haul up when I need it.
Cabling/bracing: Same as above, but add a second line for setting second system for moving 'to and fro' from one end to the other of the cable pathway, and a bucket for the tools with a second adjustable lanyard for hanging that bucket on 'anything but me', and I haul up the drill, cable-cut-to-length, dead-ends, etc on the tail of a rope. Pretty much the same for lightning protection installation, just the tools and materials change.
Takedowns: Saddle, spikes take the place of the foot and knee ascender, longer lanyard to cover add positioning, the chestie becomes a shoulder strap to better hold the weight of the large saw on my saddle when I get down to the big cuts, helmet with radio ear protectors (for coordinating with the person lowering, and managing the busier flow of material). I start with a 346XP with an 18" bar- take it with me to cut away the branches as I work my way up to where I start disassembling/lowering out the major wood. Sometimes if it's going to get big fast I'll just haul the large pulley with me, rather than haul it up later hand-over-hand. I'll do the same with a tag line if the top doesn't have a definite lean or it's a windy day.
Crane days: Saddle, spikes, lanyard, helmet with radio ear protectors, rope in a bag on my belt, for rope control, Akimbo for the quick midline change from hanging off the hook to positioned in the tree. Saw lately has been the 562XP. Definitely phone in the bag for shots while waiting for the crane slings to come back for the next pick.
Rec climbs: Depends totally on where it is, what we are climbing, whether we're just heading to the top to see the sights, or setting up a large cargo net platform for a picnic, or a zip line, or an elevator high in the tree, or getting multiple kids up in a tree for a first experience, or like we did in the "Charlie Tree" outside Atlanta, where we set multiple lines for ten climbers playing tag...