I had a tune in my head I was trying to play and it had a G followed by a C then Em but the tune goes lower than that. I am pretty new to guitars and music in general so I am trying to keep everything as simple as I can.
The lowest note in standard tuning is an E. However there is an open D tuning that is common called DADGAD (that's the notes the open strings are tuned too) which will get you an E flat on the first fret and a D on the open 6th string (lowest string). But now you will have to change the way you play all your chords.
You can play chords such as D in standard tuning but the root note (d) would be up an octave (on the open 4th string) Or C with the root on the 3rd fret of the 5th string. This may let you play your song but the bass line in your head might not be exactly as you hear it.
Another option is to transpose the whole song up to a new key. Meaning the lowest note you're hearing becomes E which is the lowest note you can play in standard tuning, and all the other notes (chords) get shifted up. Example: you are hearing d major and then e minor, change that e major and f# (f sharp) minor. It will have the same harmonic properties but now will be easier to play on the guitar.
The question has some ambiguity in it. What do you mean by E and what do you mean by "lowest"? If by lowest you mean that the lowest note in the chord is E but the others can be "high", then there are several options, Emaj, Emin, E7 etc. If you assume standard tuning (and I see many answers have mentioned other tuning) and if I assume you mean that all notes need to be the lowest pitch possible I would vote for the "lowest chord" as being E-11 in the following voicing (E, A, D, G, B, E) = (1, 11 (4), b7, b3, 5, 1). But I think you may be referring to something else. Since your question title did state "standard tuning" I am sticking to this answer.
One option to get a chord that sounds like it has a root note that's lower than the low E is to use a second inversion voicing. If you put the fifth of the chord on the bottom, it implies a root note below that. Eg. D/A, C/G, Bb/F, A/E.
See this question for more detail: Is it possible to create the illusion of a sub-harmonic?