Beyond 2nds and 3rds, of course, the intervals become larger, and perhaps can be a little trickier to identify on a staff. Fortunately, with a little practice, this becomes easier and easier, until soon enough it is second-nature.
Here's a handwritten sheet showing what different intervals (unisons up to octaves) look like on the staff (5 lines, 4 spaces) The ability to quickly recognize an interval (and eventually, several intervals/chords at once) when reading from written music notation--especially when sight-reading--is very valuable.
|I hope my handwriting is legible for you. Click to enlarge.|
The basic nugget of information to take-away from this post is this:
Intervals of Odd Numbers (1, 3, 5, 7) always appear as line-line or space-space.
Intervals of Even Numbers (2, 4, 6, 8) always appear as line-space or space-line.
To help visualize these intervals in another way (pianists especially), it would be a good idea to plunk out each of the intervals depicted above on a keyboard. Begin doing this in the key of C major (all white keys), and when comfortable try out some other keys in order to incorporate black keys into your visual memory bank.