The "first class" train that leaves early in the morning and returns in the evening from Delhi to Agra is very safe and is worth taking for the experience. I took it to Agra and was driven back, which is not a bad way to go as you can leave Agra when you want on your own schedule.

I have been to both Delhi and Mumbai and would choose Mumbai over Delhi as a tourist destination. I haven't been to Goa, but if you want to cut that a bit short you could easily add a couple days in Mumbai as they are fairly close.

My biggest recommendation is to avoid taxis and arrange for a car and driver for the days you want to go sightseeing. For 8 hours/80KM it shouldn't cost you much more than $40 US.