Be careful on the lino. If the house was built before the late 70's, it potentially contains asbestos. Same deal on drywall actually.
In a place like that, I think you'd want to get rid of every lick of drywall and start fresh. If you are going diy, drywall is a bitch. The actual boarding isn't too bad, and the tools are cheap to do it, but mudding is an artform.
I usually do my own drywall, but am far from a "drywaller" and in the end, I'm so-so. When I bring my drywallers in on larger projects, and watch them, I feel even worse about my skills.
Installing a bathtub is actually really easy.
Hire out the plumbing.
Installing a toilet is also really easy.
That leaves you with 1)tiling 2) drywall and whatever you do for flooring
The best thing you can do is do all your own demo, and hire out parts of the project. A drywaller that works in your area can fit you in around other large projects and if you aren't in a hurry, its kind of free money for them. Drywall works on volume, as there is downtime on each project. You end up having a day on site where all you do is come and slam some mud on the wall and leave.
Tilers expect to get paid large for their efforts.
You should totally be able to paint it yourself. Use tape if you can't cut a straight line.
Then its finishing baseboards, the vanity and so on. Surf craigslist for a guy that has nothing to do for a day.
Boom! I just saved you $3000. I'll take 10% for my efforts
