The hande on the toilet was not flushing without lots of pushing. Now the handle had broken but I can flush the toilet by pulling up the thing inside the tank. Why is it sticking or so hard to flush?

The handle passes through the tank wall and may just be corroded or rusty. You can replace that item at most hardware stores and box stores for less than $6. YouTube - ‪Toilet Handle Repair‬‏ (http://www.youtube.com/watch?v=ZrnJDM3Dc4w)

I was actually pulling up the stick connected to the handle inside the toilet to flush so that I would not break the handle and it was still very difficult to pull up so the toilet would flush. My husband broke the handle because he just kept pushing it. Could there be too much suction or sticking in tube thing in the toilet that pulls up with the handle?

The handle fits through a square hold in the tank. The part of the handle that fits in the square hole is also square. The fact that the handle shaft and the hole are square is what keeps the handle lever horizontal. However it fits rather loosely. When the nylon shaft gets worn the square shaft can turn and get get out of position.
Handles are not expensive and easily replaced. Please note that the handle and the nut that holds it in place have a left hand thread. In other words you have to turn it opposite of what you normally would to remove.

Could there be too much suction or sticking in tube thing in the toilet that pulls up with the handle?

No. The lift rod (what you called a stick) is hard to lift because the square part of the handle has gotten catty cornered in the square hole and squeezing down on the rod where it passes through..

Over, the nut for the new handle will be left hand threads. Good luck

Also note that when I have a customer with a sticky or hard to flush toilet it is usually the FLAPPER that needs to be replaced.

Replace the handle and at the same time, replace the flapper... should resolve both issues!

Mark