If I am holding Grove of the Garden with Blue Sentinel, do I draw 2 cards extra?

Yes, you draw 2 cards total — that's 1 extra card beyond the normal 1.

When you hold the battlefield where Blue Sentinel is at Grove of the God-Willow, Blue Sentinel's passive causes all your hold effects at that battlefield to trigger an additional time. Grove's "When you hold here, draw 1" triggers normally once, then a second time, for a total of 2 draws. You do not draw 3 cards (the extra trigger gives +1 additional draw, not +2).

Blue Sentinel also makes its own "When I hold, Add [A]" trigger twice, giving you 2 Power at the start of your next Main Phase.