I have typically seen "suchen" and "suche"
but have also ran into "schnellsuche".
Is there a time when you would use one over the other? and what are some of the differences in how formal they are?
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It's merely a matter of preference. However, Schnellsuche means "quick search."
It's hard for me to tell whether in the second example, they mean to use the imperative form of "suchen" or the noun "Suche." I would guess the noun. Additionally, should I be making a website in German, I would use "suchen" rather than "Suche".
The word Schnellsuche implies that there is (in addition to this "quick search") also a more detailed search function, which allows you to fine-tune your search request, as opposed to the quick search, which performs the search based on default settings.
Suche can be both the noun, but also the imperative of the verb ("Search now!"). Both is spelled with a capital "S" here because it is (sort of) the beginning of the sentence. Having the ellipsis suggests that the noun is meant.
You could also use Suchbegriffe, which translates to search terms, but most common are Suchen or Schnellsuche (if – as I said – there's a detailed search, too).