I've lived in Delhi and visited Agra, Amritsar, Jaipur, Darjeeling Ranthambore and then some areas further south while I was there.
One of my favourites was Amritsar to see the Golden Temple then head out to the border to watch the Wagah-Attari border ceremony. Definately rated highly.
I also loved Darjeeling for the walking and views of the Himalayan range.
I realise neither of those are on your list and Darjeeling is not really North. But I mentioned them because I loved them.
I also loved the Taj Mahal - even though everyone in India rolled their eyes and told me it was over rated (they all ecommended the Golden Temple instead). But I thought it was stunning so I am really glad I went.
I stayed in Ranthambore but it's obviously right next door to Sawai Madhopur. We took a great safari ride through the park and got some great sightings of wild tigers which is a brilliant memory.
Because of a inherited health condition, my GP sent me out with a stock of antibiotics to be taken if I got an upset tummy. The first time it happened I held out before taking them and was very ill. Thereafter, I took the antibiotics straight away and it saw off the bug very quickly. If you can't take any with you then don't hesitate to go straight to professional medical care if you get ill to the point of regular stomach and toilet trouble. Don't try and tough it out. Take immodium with you: but remember this just delays the inevitable so only worth using if you have a journey ahead of you and find yourself with the runs at the wrong time. If it's not getting to crude now, I received some great advice from a long-termer before I arrived: never, ever, ever trust a fart. I'm not joking: it really was good advice.
For the most part, our stomach bugs could be traced back to: food served at a nice/respectable hotel that was not quite cooked right. For each of us we were suspicious about the food but ate it anyway, believing the hotel to know better. The lesson here is: if it ain't piping hot and clearly cooked through - send it right back immediately.
Even with that inherited condition, I did not take malarie tablets having talked it over with my GP ahead of time. I did have almost every vaccination going, though.
My advice would be to either prebook your travel and guides with a decent firm or be on the clear look out for scams. They are often fairly harmless, such as a suggestion to go to a particularly good clothes shop where the hard sell is then applied etc. You can avoid them with a clear head and being prepared to say NO very clearly, if it's something you don't want - but if this makes you feel uncomfortable then book with a guide/driver/etc you know is not going to try this.
Lots of what you'll see which be historical so I think it also helps to have a basic history knowledge of the country to help everything 'fit' in your mind.
And enjoy!