There are two types...

There are two types of people - those who put people into two types, and those who don't. A similar, but much more interesting statement along the same lines applies to music.  There are two types of pieces - ones which start with a pick-up and one which do not. Of course, this is a true statement, but you might also ask yourself, what's the point?
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.

Rubato
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.

Phrasing
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. 

Examples
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. 

I am not saying that time needs to be taken every time these situations occur.  Quite the contrary: I believe that pianists of all ages and abilities tend to take too many liberties. Our composers were really good.  They don't need a lot of our "help".  Rubato is an important tool.  Knowing where to use it and knowing how judiciously to use it, is a true art form.  Applying the ideas presented here will help with precisely this.