Recognizing what kind of a piece it is, is essential to interpreting a piece, to knowing where to employ rubato, and to thoroughly understanding the phrase structure.
Before I get into more detail, one bit of clarification: just because a piece is fundamentally a non-pick-up piece does not mean that one will never find a pick-up (and vica versa). Pieces ARE fundamentally one or the other and that can be figured out at the beginning of the piece. But again, the composer can of course explore the opposites in various places/sections in the given piece.
If you are dealing with pick-ups, avoid taking time between the pick-up note(s) and the downbeat. By all means explore taking time AFTER the downbeat.
If you are not dealing with pick-ups, take time before the downbeat and not after.
Very rarely should one take time both before and after a downbeat.
The phrase structure will be informed by being aware of pick-ups or lack thereof. The composer will not always denote the phrasing in their scores. So pay attention to whether the music moves towards a beat or from a beat. This has huge musical ramifications and even has technical importance as well. Muscular relaxation will change depending on where phrases start and end.
Let's look at one of my favorite teaching pieces, Chopin's a minor waltz op. posth. The pick-up to measure 1 should be strictly in tempo, moving towards the downbeat. Time can (if desired/necessary) can be taken to assist the LH getting from beat one to two, and the same can be said for between beats 2 and 3. Beat three of the first measure is again a pick-up, even though not notated here. Therefore the young pianist should avoid taking time between the second a-minor chord and the downbeat of measure two, even though it would be more comfortable.
In measure three the melodic notes A-G-F should be considered pick-ups.
The rest in the LH in measure 8 needs to be adhered to, thus allowing the E in the RH to belong to the following measure 9. A clear pedal change on beat three is absolutely necessary.
In measures 17-24, Chopin has a short excursion into a non pick-up section. This helps support the notion of a more sighing character and technically assists the pianist by allowing some space between the repeated notes across the bar lines. This make the ornaments easier to execute.