Since bigxxx covered most of the tourist related information, I will focus on the women
Honestly speaking, your best chance of doing any damage on the girls front is in Goa. Apart from Goa, I doubt you are going to make much headway anywhere else in India within 3 weeks. I would say it is going to hard even if you are holed up in one place for the entire 3 weeks, so with traveling around it will come out to nearly impossible.
Mumbai probably has the best nightlife in the country after Goa, but that does not say much since the entire country simply sucks when it comes to nightlife. Some people do like it, but most of those people don't know what good night life is. After Mumbai, your best bet at having a decent night life would be in Bangalore, yes everyone you ask will tell you about the famed "Pub City" of India, but don't get your hopes too high. But compared to the rest of the country, it is better in Bangalore I guess to a small extent.
Since you are only visiting for a short time, day game/online game isn't going to help much either. India is just not the place where you can game women easily and get them back to your pad within a week, leave alone on the same day/night. What you think of game can be easily misconstrued as harassment/teasing women in public and can lead to frowns and trouble if you are not careful.
If you are very lucky, you might get to meet some cool broad minded women out during the night (very unlikely), but even in that case tread carefully in terms of trying to game them. There is another post called "India - Chances of getting laid?" or something along those lines, where I posted a longer post, check that out to get an idea of how things work here.
Mumbai & Goa are probably the best places in the country if you are going after women, and even in those two places targeting western women/european holidaying crowd/non-indians is the only viable option. Indian women are not receptive to talking to strangers, and Indian men mostly will come off as cock-blockers like no other.
Mumbai also has this advantage since a lot of film crew extras tend to be there, and they might be open for some gaming but most of these ladies will only be seen around in high end, upscale places normally. Generally speaking unless you go to the posh and fancy places, the quality of women won't be that great. But in the high end, nicer places you will definitely come across some beauties.
Don't know how you look, you style etc - but if you are white and dress reasonably well you will be at an advantage. Since places tend to treat foreigners well for the financial gains involved.
If you decide to visit Bangalore, drop me a PM and I can provide further information regarding the best places and perhaps some contacts.