Important tips about joining Salāt:
1. If you enter the masjid and found an individual praying:(a.) Do not ask him what Salāt he's praying (You can join anyone in Salāt whether or not he's praying the same Salāt you intend to pray).(b.) Stand next to him to his right.(c) . You don't need to do iqāmah, just say your Allāhu Akbar.
2. If you enter the masjid and found 2 persons praying:(a.) Do not push or pull anyone, just stand next to the man to Imām's right. The row will arrange itself and take proper shape (either the Imām takes a few steps forward, or the one praying with him takes a few steps backward together with you).
3. If two or more of you enter the masjid and found a person praying:(a.) Do not touch him to notify him of your presence(b.) Form a row behind him and make your takbīrat loud enough for him to know
4. If you enter the masjid and found the individual or congregation on the final tashahhud(a.) Join immediately and make the tashahhud with them.(b.) After their taslīm, rise to complete your Salāt individually, or, more preferably, start your own Jamā'ah with a fresh iqāmah.
5. If you enter the masjid and found someone who rises to complete his Salāt after the Imām's taslīm, join him for another congregation (my preferred position among the two views held by scholars).
I hope this helps.