Check the Mechanicals
You will want to pull and grease the wheels, check and tighten the nuts and bolts over the whole vehicle to keep it safe. Moisture can collect in the hubs, no grease, so you tear up the axle or lose bearings. Loose bolts can finish working loose, make big wallered out (technical term!) holes so parts give way while being used!!
Just tipping the cart up to view and test the nuts on bolts for tightness, is a big step forward in being safe.
Just guessing, but cart is probably an inexpensive metal one, bike tires like in many sale catalogs? They are very cheaply made, so you really want to look at things close to prevent problems. You can easily tighten things down as you look them over. Wheels come off fairly easy, need the hubs and bearings to have LOTS of grease to prevent metal wearing down as it rubs together. Check air in tires, should give you numbers on the sides of tire for inflation level.
Go over your harness, add some conditioner if leather, to keep it soft and pliable to buckle things. Check all folds for wear, cracks, dry, probably need to be replaced before they break. I use leather conditioners, NEVER use Neatsfoot oil because it is so harsh on leather. NOT what your Grampa used to use!
Synthetic harness needs you to open and shut each buckle, check for wear in the folds, before using. All folded places need checking for wear that can break on you. Worn places need attention from the harness repair person, maybe replacement to prevent breaking. Usually cleaned easily with just some Ivory liquid dish soap, soft brush on dirt, with a good rinse to remove soap. Hang to dry. NO conditioners, which will actually damage many of the synthetic products.
I do not recommend using any snaps on harness, they are cast metal and often break under very little pull. All harness parts should be removed from cart after each use, so you TOUCH them to feel any wear points.
This total checkover of harness and cart is recommended each year or if you haven't used them for a while. You can't just put them away and expect them to stay greasy, usable with no attention over time. Leather in particular is subject to breakage from wear, mouse chewing, dirt wear on stitching, so you need to look it over closely. One part failing, can lead to other parts breaking in stress, so then there is a "Driving accident" with your equine, that didn't need to happen.
Be safe by going over things before hitching, so your driving time is ALL fun.
Sorry, no help with rust and paint choices to get the shine back on. Cordless drill and wire wheel for rust sounds good. You DO want to wear some kind of mask and safety glasses when removing paint so you don't breathe it in. Same with painting new stuff on. Even with new paints, the tiny pieces dry or new paint drops will get in the air to get breathed in. That air laden stuff gets into your lungs to coat it. Doesn't work out easily. Prevent the problem with a mask for removal and painting times.