The hand thing could be poor bike fit, if you're putting a lot of weight through your wrists, it could just be a weight thing and will get better as you get lighter. It can be helped with better grips or padded gloves. There's a nerve down the middle of the palm that gets squashed which is a big cause of it. Getting some blood back in to it and changing your hand position around (on a hybrid easier with bar ends) can help. Ergo grips and bar ends helped when my wife was riding flat bars, she is putting a lot of weight through her wrists.
Gears, could be 'bedding in' especially with cheaper components. There are screws and things that adjusting helps matters. The bike shop is a good idea as they've offered but if you want to get in to it yourself it's easy enough to find how on youtube although a workshop stand makes life a lot easier (if you haven't realised, spending more money on cycling is a doddle!)
I found bicycles just scarily thin when I first started riding them again because I was used to motorbikes, plus at 22 stone it felt like balancing bowling ball on a kebab skewer. It soon passes. With two wheels, the faster you are going the more stable it feels (up to a point).
Work up the muscles over time, pushing on is a good thing, damaging stuff so you have to stop for a while is a bad thing. It'll come sooner than you think providing you keep cycling. Get in to the habit of stretching at the end of the ride, it's one I wish I'd cultivated.
I use a mix of Map My Ride and Strava. Both have their positives and negatives. If you have a smart phone then there are plenty of free Apps and it should have plenty of battery to record a short ride. You can do it for nothing if you keep reasonable records (both will allow you to map the route you are taking and you can put in a start and finish time which will give you an idea of whether you are getting quicker) but something that logs the ride (higher end bike computer or Garmin of some kind) makes life easier and gives better data. That's pretty much the thing with cycling, you pick your point on the line between hassle and expenditure :-)
I spent over a year just logging distance from a basic bike computer in to an online log and using that to compare things. Now I upload from my Garmin Touring + which is a lot easier, but I needed to convince myself I wanted to spend the money. Knowing you're serious about it helps that :-) .