Tiger's Modern

The idea of b7-b5 is to pressure White's center (mainly the e4 square) with moves like Bb7 and/or b5-b4.   This usually works best against 1.e4, 2.d4, 3.Nc3 setups for White, as the Nc3 is a nice target for Black's b-pawn.

However, 1.d4 players very often play 2.c4, which makes an effective b7-b5 move for Black harder to achieve,, as it often just loses a pawn.  So Black usually tries other ideas, like a quick e7-e5 (1...g6, 2...Bg7, 3...d6, 4...e7-e5, etc.)  This is actually the system that Tiger recommends against 1.d4 in his second book.

This is very helpful ARenko! Thank you. Ooh, I have the Tiger's Modern book, and it is very good. I didn't know he'd written a second. Would you recommend it?😀

check this opening chapter about c4 moves in modern. (Variations based on book "Modernized Modern Defence")

If you have the 2005 version of Tiger's Modern, the 2015 version is probably more up-to-date.   I only have the more recent version, so I can't compare the two, but the 2005 version got very good reviews and I suspect it would be fine to start with that.

@ARenko basically summarized that very well

If white plays 1. d4 and 2. c4 against 1. g6, I line I like is this Sniper variation where black gives up the bishop to create pawn weaknesses

of course, you could also just play 4. f5 and keep the bishop instead. All preference

As previously answered, the c-pawn would prevent the b7-b5 push. It is still possible but would result in a loss of a pawn for an open file which is not worth it in that specific situation. When they do play c4, I love to play the averbakh system and end up in a KID position. If you don't want to learn all that theory there are other alternatives like what @DasBurner mentioned. I am not going to call myself an expert, or even remotely good compared to other people, but if you need any help on the modern, let me know. One of my fav opening to play.

I will just second Gibbilo's comment: Cyrus Lakdawala played the modern for many years and he does an excellent job of explaining the ideas behind the Modern in his books.

There is a good club to join the Modern with plenty of vote chess games going on

I used to play the a6 modern years ago and mostly followed Nigel Davies' repertoire. But if White goes for the big King's Indian center with e4/d4/c4, insisting on the Modern move order just ended with Black playing a KID with less options, so I finally went for a KID directly after d4/c4. One of the reasons I stopped playing the Modern.

Thank you very much for all these very helpful comments and ideas. 😀👍