Good for you to be in India!
Rishikesh is nice, but the area around Lakshman Jhula is a bit too touristy for my taste.
When I used to go there with yoga groups, I stayed in Shivananda Ashram (it is in the Ram Jhula area), but I think you would not be able to stay there as an individual tourist.
My second favorite place to stay is Ved Niketan, on the Swarg Ashram side of the Ganga. I haven´t been there since they renovated the rooms, I guess prices went up, but it´s a quiet place, away from the crowds.
About your planning to go to Gangotri:
The official opening date for Gangotri temple is April 26 (tentative date), so I´m not sure if you can go to the region earlier, sometimes there is still too much snow earlier in the year, or the police won´t let foreigners pass on the road to Gangotri, many guesthouses will also still be closed before the official date.
And once it is officially open, you will be on your way with thousands of Indians, which is an experience in its own right.
Couple of years ago I went there with my girlfriend to do the Chardham Yatra, so we went to Yamunotri, Gangotri and Kedarnath, but after Kedarnath we were so fed up with the crowds and the ugliness(rubbish, the inconsiderateness of the horsemen and their maltreatment of the packhorses) that we did not go to Badrinath anymore, but went back to Rishikesh, which was like a safe haven for my girlfriend.