It's impossible to answer because we have no idea what skills you already have.
I've fixed several plumbing issues myself just by watching YouTube videos, but it's often taken hours, especially with an older house with.nonstandard plumbing fixtures, and place is only 50 years old.
I've had to hire a plumber for a few jobs that I just could not do myself because my plumbing skills are limited. However, you would be surprised how much you can get done with even very, very limited plumbing skills. Again, I have no idea what your existing skills level is, so that's really all I can say.
I totally agree with pp to leave pretty much anything cosmetic alone for awhile. For one, it doesn't need to be done, and two, I like to live on a place for awhile before finalizing what I want it to look like. I had a lot of ideas of what I wanted to change when I first moved in here, but over two years, I've changed my mind on a lot of them.
Take your time, get to know your house first before making permanent, cosmetic changes. However, the best money saver will be to keep as much original as possible. My kitchen and bathroom were hideously dated when I bought this place, but some paint, refinishing cabinets, and new cabinet hardware was all it really needed to be updated. Well, I could use a new kitchen countertop, but I can't decide on colour, so that will probably take another year of contemplation before I commit to more change. I *just* dramatically changed the cabinet colour, so I'm getting used to that first.
I also agree to try cleaning the carpet first, unless it's truly irreparably bad. If it really does need replacing, then definitely tear it up yourself, this is an easy DIY job. I can't say much else about flirting because you haven't said what you want. Most people these days seem to go with vinyl plank flooring, which is very DIY-able. I've never done carpet myself, so I can't speak to how easy or difficult that it, but intuitively, it seems doable, but frustrating. Then again, you never know what you will find under carpet, in a house that age, it could be hardwood and you could be in a refinishing situation, which is doable DIY, but not pleasant if you live in the house.
As for the roof, sorry, I have no input on that, I've always lived on condos, so never needed to make roof decisions, but tons of people here can give you input.