Hmmm...
More likely than the pump is the coupling; whatever that part is called... (not every brand has this part; you didn't mention the brand in the OP)
Tip the washer back and look under it... if you see bits of shaved black rubber, it's that piece... about $5-10, and 30-45 minutes to replace (the first time) If you should ever have to replace it again, it's a 5 minute job once you've learned how. Hmmm... darn it.. now I'm double questioning myself if that part stops it from draining too... I know it stops the agitating... I think it stops both, but I could be mistaken.
For what it's worth, the most common problems with washers and dryers are easily d-i-y projects.
Like someone said above, it's like legos - take it apart til you find the broken piece/problem, put it back together. There are a lot of sites online with instructions as to how exactly to take it apart.