I am fascinated by all this. This is a good plan, and will help you get fitter and happier. My biggest question mark is your current bike. I ask because not buying a new bike, and not doing much to it is by far the cheapest solution. So far I have not yet seen any facts that argue against this route. At the moment you really do not know much about cycling or about bikes. So you cannot judge what you will need or what you will like. So unless what you have does not really qualify as rideable, use the existing bike. Get yourself a decent bike maintenance manual like the Park Tools book, and spend a weekend or so on an overhaul, replacing cables, brake pads, greasing all the various bearings etc. Check if it needs a new cassette (the chainwheels at the rear), but do not yet spend that money. Just ascertain there is no fatal problem anywhere with the bike. Then ride the bike for a few months. After that, and if it still survives, come back and ask about camping gear. Buy quality and really light camping gear, tour for your summer holiday, and after that, come back and ask for advicew about a bike. Unless your bike is beyond repair, of course. But you have not yet convinced me.
Willem