What steps to take to replace bike frame
There are many issues for the new frame:


Fork drop out (front wheel spacing) might be wider, 100 mm for most modern bike
Frame dropout (rear wheel spacing) might wider, 135 for most modern bike
Headset likely to be wider, so you have to buy new headset with compatibility to a 1" fork (I guess?), 1-1/8 inch to 1-1/2 inch for modern bike
Bottom bracket is likely to be ok
Front derailleur clamp may be too small
Seat diameter might be different 


Rules of thumb is that if the bike has no sentimental values, it is always more economical to use that money and buy a second-hand bike. Unless you would like to spend extra money to tinker and have fun with the bike in your own time.