there are many ways to get a chain but basically, the main way to do a chain is when you delete a number of blocks the blocks on top fall. be for they fall all the way switch them around so when it falls it will delete some more blocks by getting 3 or more in a row.
To unlock the Advanced Puzzles you need to clear 60 Basic Puzzles. If you use a hint, the puzzle won't count as cleared, but you can always go back and try again without the hints. A star will appear next to each puzzle that has been cleared.