I presently have 10,20 and 30 amp fuses. Last night the entire upstairs went out. If I replace the old fuses with push buttom fuses will I be able to go from 10 to 15 amps and so on...

Absolutely NOT!!

The fuses are there to protect the cables. A cable listed for 15 amps cannot take more than 15 amps for an extended time. It will start to char/melt and then the conductors will become exposed and here comes the fire.

What is the response time of your local Fire Dept? Doing what you want will give you the answer.
If you are still on a fuse box and blow them, you are overloaded. Most houses today need at 150 Amp service or more, both of our houses are 200 Amp. NEC 230.79c 100 amp minimum. You need a qualified electrician to do some upgrading. With a question such as yours, I do not recommend a DIY project.