An encounter of normal difficulty for a group of 3 level 1 characters should have around 300 xp worth of monsters in it. A hard encounter for the same group could have 450 to 525 xp worth of monsters. Your problem with HS1: The Slaying Stone is that the encounters are balanced for a group of 5 players, and thus have at least 500 xp worth of monsters in them.
For example Encounter 1 has 2 gray wolves (125 xp each) and 3 ravenous wolves (100 xp each). If you play that as written, that would be very hard for your players. In consequence they would earn 550 xp divided by 3 or 183 xp each. At that rate they'll reach level 2 after only 6 encounters, if they survive that long.
What you probably would want to do for a group of 3 players is to remove some monsters from the encounters. For example use only 1 gray wolf and 2 ravenous wolves, for a total of 325 xp, a standard difficulty encounter for 3 level 1 characters, giving each of them 108 xp.
Note that even with the reduced monsters and xp your group will be at least level 2 at the end of HS1 (the extra encounters are there only in case they haven't made level 2 because they skipped some encounters). Your next adventure, H1: Keep on the Shadowfell is made for 5 level 1 characters. It will probably be all right to play it with level 2 characters, but you might still need to take out a few monsters from the encounters, because for 3 level 2 characters a standard encounter is 375 xp, while for 5 level 1 characters it is 500 xp.