As far as the joint injections - it really depends on what you're putting in there if you're wondering if it's a bad idea. Anything steroidal or anti-inflammatory that gets put in there is actually going to degrade the join gradually and will only mask the issue until the horse is completely lame. What you want is Hyarounic Acid (brand name Legend) which is actually synthetic joint fluid and is going to help the body provide better cushioning in the joint.
As far as the joint supplements, I am aware that some people swear by them but clinically they do next to nothing. Your best bet is injecting (IM) Adequan which is going to slow degeneration of the cartilage in the joint and even improve its condition. Adequan is clinically proven to do this. And if you're going to go this route you also might try IV Legend and scrap the joint injections of it. Both of these injections monthly are the best recommended thing for arthritis.
Another thing that is actually recommended more highly then the above injection is daily bute or banamine, the only problem is that it is not permitted for show horses to have these medications so if you're showing it cancels out this option, but for a pleasure horse this is definitely the way to go. You may also want to occasionally do some IV Legend, but you don't really need joint injections or oral joint supplements, but if either or both make you feel comfy then go for it. The joint injections can be slightly dangerous though (risk of infection, risk of damage to structures, stall rest required, etc..). Also - this is a horse that needs 24/7 turnout in a large field with hay and water separated to encourage him to walk a lot, ideally being on grass and grazing all day.
As far as having an arthritic horse and developing a training schedule, it's going to be a good month before you even think about cantering him. And as far as lunging try to avoid it as much as possible. Continually pounding around a circle wreaks havoc on even healthy unfit joints. For the first few weeks (6 days/week), hand walk him on a good surface (arena, track, etc) that is smooth and not too hard or soft for half an hour to start and gradually increase to one hour. And make it a brisk walk too. Then if you must, hand walk for 30 minutes, trot for 5 on a lunge, hand walk for 30 minutes. Ideally he should be ridden to avoid the lunging and this should be repeated 6 days/week for a week to two weeks. Then you can extremely gradually increase the trot and ideally be doing his work outs under saddle. The first time you canter should be 6 weeks into the program and only for 1-2 minutes, then again gradually increase.
In 8-10 weeks you should be doing 15 minutes walk to warm up, 30 minutes of "dressage" work including basically everything from first level (if his brain can handle it) and probably about 5-10 minutes worth of walk breaks followed by 15 minutes of walk cool down.
Up until now, work outs should be limited to 60 minutes, but as soon as he can comfortably perform above work out you can start to work him slowly harder, but keep the walk warm up/cool down periods 15 mins each. He needs this time, and especially if you have him in a small area. At this point I would for sure begin lessons as well.
After work outs it would also be advisable to cold hose or ice his hocks for 15-20 minutes.