I am trying to create a survey result sheet in which there are 2 cells
Cell 1 : Input - Any number as described below
Cell 2 : Output : Gives out value of 1,2 or 3 in accordance with
input.
Less than 2 years old : 1 pt
2yrs 1 month to 5 years : 2 pts
Greater than 5 years 1 month : 3 pts

If the input cell is A1, one way is to put this in the output cell:
=(A15)+(A12)+1

If you insist on using "IF" another way is:
=IF(A15,3,IF(A12,2,1))

As others have noticed, the conditions in the original post have one-month
gaps (after 2 yrs an after 5 yrs). The above formulas assume that the gaps
really weren't intended. If that's wrong, modify accordingly.