I'm trying to intonate my Tele, and the strings on it are not what you'd call a new strings. I'm wondering (because I'm having troubles with the first few frets being sharp), does it matter a lot if the strings are not new? Or should you definitely get a new set of strings before you do any action on the nut, truss rod, bridge...
Yes - old strings can make the strings intonate significantly flat, so it's a waste of time setting the intonation with old strings, because it'll be completely wrong when you replace them.
Definitely. You should always check the intonation and setup when changing strings. And you should always change strings if the intonation and setup seem way off.
You do always want to intonate for the strings that are on the guitar, so getting a new set. However, if it's a 3-saddle style bridge, you might not be able to get it perfect as they aren't the easier to work with