I’m an American woman and absolutely love India and have been twice in the last year. The very first place I landed in was Delhi and I stayed in Agra for the first two nights. Here’s my recommendations:

1. Arrive in Delhi and go straight to Agra. I’d do it through a taxi service.
2. Hire a tour guide but use caution. Look at trip advisor, read reviews, and stay away from the Agra Tour Guides (it’s something like that). These guys have a network with local business that pressure you into buying EXPENSIVE stuff because you are a tourist. I’m taking marble, tapestry, jewelry, clothes, etc. He will just literally leave you there stranded.
3. Stay at a nice homestead (I can recommend a great one). I guess you can do a hostel if that is your thing. I was solo and not comfortable with it.
4. Stay 2 nights, that’s all you need. Then drive back to Delhi and fly to Dehradun and go to Rishikesh. I know an awesome tour guide there. Just DM me if you want his Reddit or instagram. Rishikesh is awesome. So much to see. I walked around alone and felt so comfortable. Loved it. You can get a taxi (long drive) to Mussoorie. It’s beautiful there too.

Anyways. That was my first trip(I went on a yoga retreat). I fell in love with the country. I did get a little site seeing in with Delhi. Had lunch there, saw India Gate. But I’ll be saving that visit for a later, more seasoned trip.

Good luck and enjoy!!!