I guess 2-3 days will not be enough cover more than 2 places. Agra, Lucknow & Varanasi are the best places to see however, Agra will take one whole day and in Varanasi you need to have 2 days in order to cover the Fort & Sarnath as well.
Ohh I forgot about precautions, Be friendly but don't share your details like where you are staying and alone. Keep valuables safe, if purchasing anything, bargain. For commuting try booking through Ola, Uber or autos. Happy Travels!
You can easily cover main places
Day 1 Reach Agra and you enjoy
1 Taj Mahal
2 Agra Fort
by evening you will be able to cover all these monuments
From sikandra you will get direct bus to mathura
in night you can go to temples
iscon temple Day 2
Enjoy your holi in mathura (Barsana) Day 3
you can reach delhi (if time permits than you can also enjoy delhi places ) and get back to your home
It's best destination for holi...you should go to Varanasi... it's a magical city, you will that you are in lap ao lord shiva, feel the culture of Banaras it's world oldest city, must try bhaang, Pan on holi