Dripping toilet with vertical plunger
It sounds like you have 2 problems:


"constantly filling"- the fill valve (often referred to as the ballcock valve) needs repair. Often this is simple, get as much information on the valve make and model (often there are maker's marks on the valve itself) and buy a rebuild kit at a good local plumbing shop.
"water continues to drain out"- the flush valve (this is whatever plunger/ball/flapper type sealing apparatus that is raised/lifted/moved when you press the flush button/lever) needs repair or replacement, same advice for that as in prior step.


Note- if you were to include pictures of the internal tank components we might be able to give more information.