First of all, pull the quick release/safety release (red cord, usually) hanging from the "traveler" (the part that goes along the rail/tube, what the door is attached to).......try the door by itself.
You should be able to open & close the door easily, without much effort. (Keep in mind, if you are Ah-nold S., I mean easily for the avg. guy.
If you can't, start looking for problems with the door....bent rails, bad rollers, bad bracketry, bent door panel, bad springs, misadjusted springs, a wooden door that has absorbed water....etc. Sounds like you've kind of done this, but, make sure the door is eliminated. About 1/2-2/3 of the time, it's the door, not the opener. Especially on older doors.
Also, remember that, for the most part, the "opener" is not intended to be an opener...but rather a guider or starter. It's not intended to pull all, or even more than a little, of the actual weight of the door.
If you are satisfied that the door is o-tay, then look at the opener itself. Cycle it without the weight of the door. Does it cycle completely, both up & down?
Then, reconnect the opener to the door via the quick release, & try the wall button. Does it work then? Hold the button DOWN, without letting up. Does it work OK if you hold it down, but mess up when you just push & release? In that case the beam sensors may be mis-aligned, or broken, if the GDO has them...used since around '93. Not fixed yet? Next....
Without getting real specific, there are generally four things to adjust on a GDO: opening force; closing force; open limit (i.e. stopping point); close limit. Different brands have different styles & locations for these, & older openers have (usually) only one force (sensitivity) adjustment.
It sounds as if you have a problem with insufficient opening force. There generally is a white or black knob (actually, two knobs) on the back of newer openers, one for open, one for close. Find the open force knob & turn it up about 1/8 turn...give it a try. If no luck still, try another 1/8. Keep going if needed.
Older openers have (usually) one large knob on the bottom, as mentioned. You can adjust this about 1/8-1/4 at a time...should be arrows on the case to show you which direction. KEEP IN MIND THAT YOU ARE ADJUSTING BOTH OPEN & CLOSE FORCE WITH THIS TYPE OF OPENER...it's basically a spring clutch.
Assuming this works, that's fine....but, ask yourself why this happened. Did something change on the door that it now requires more help? Did you have to crank the open force up a LOT? Sometimes the seasonal change requires a little bit of fine-tuning, no big deal.
Not trying to over-think this....but, a garage door is one big-***, heavy, moving object....and they can hurt people, ESPECIALLY KIDS. Most people don't think much about it...and if they have a problem, just crank the force up a little more. Or a lot more. Doing this on open force is one thing....doing it on CLOSE FORCE is a whole 'nother thing....
If it turns out to be a spring problem, as a rule it's better to let a professional handle it, unless you understand what you are doing....ESPECIALLY with torsion springs. I know of at least one fatality where an amateur decided to fix it himself.
I think "zephyrbob" here does GDO installations...search if needed.