Yep; two types will mean that a Pokemon's weaknesses/ strengths are different.
Link - Pokemon type chart
This is a the type chart. So lets use the example you wrote, Ground/ Flying. There are Pokemon with this type combination: Gliscor and Landorus.
Looking a the type chart, Ice is x2 to both Ground and Flying, which means for a Pokemon like Gliscor, Ice moves will do x4 damage.
Looking at other weaknesses:
- Rock is only x1, because it's x1/2 against Ground and x2 against Flying.
- Grass is only x1, because it's x1/2 against Flying and x2 against Ground.
- Electric is x0, because Ground is immune to Electric attacks. Immunities granted by one of the two types will override any weaknesses the other type has
- Water is x2, because it's x2 against Ground and x1 against Flying.
Type weaknesses and resistances are "evened out" or "stack" in dual type Pokemon. All you need to do is multiple the numbers together, and you will get their resistances/ weaknesses. E.g., Ice is x2 against Flying, x2 against Ground, 2x2=4, Gliscor has a x4 weakness against Ice.
As for strengths against other Pokemon, e.g. since Gliscor can learn Flying type moves, it can use a move that is supereffective against Grass Pokemon, a type that it would be weak to without its Flying type. The same applies to Rock and Electric Pokemon.