Do I simply play both hands legato? Does the slur mean that anything that falls beneath it for both hands are to be played legato? I thought I knew how legato worked, but I'm not sure anymore.
This is a phrasing slur spanning the melody line (which has its first note in the left hand and is played mf in contrast to the accompaniment played p). You phrase the melody as such legato and mf. That its first note is played with the left hand is a detail not relevant to the listener.
The phrasing slur does not concern the accompaniment starting at the full bar.
The "slur" is not a slur - it is called a phrase mark. They look similar, but a phrase mark is best thought of as how you would phrase it if you sang it. Slurs should always be REALLY clear as to which notes are slurred together. Note that not all notes under a phrase mark should be slurred. You may interpret and present the melody as you prefer it! (Slurring a repeated note would be considered 'odd', and the first bar in the RH is all on one repeated note)
And the reason the fingering is as it is, I believe, is because if the pianist positions their hands with both on a C-maj triad (RH thumb on middle C, LH thumb on G below) then the whole piece (certainly the melody) can be played without having to relocate or extend the hands - so the novice can focus on JUST the fingers.
The first note of the song, an anacrucis, could have been written as a G with a couple of leger lines, under the treble staff, and thus played with r.h. It's slightly confusing as it has been written on the bass clef, making the slur travel from one stave to the other.
As such, yes, it's played legato - if you were singing the words, they'd be legato, wouldn't they? All legato means is played smoothly and joined up together, as 'in one breath', so to speak. And, slurs (and phrase marks) are relevant to the notes under them, in that particular clef - which here is a little confusing, and didn't need to be.
As far as the triad is concerned, that's going to come out legato, as a single chord held for two bars - can't be helped!
This is obviously music aimed at a fairly elementary student. A decision was made to give priority to maintaining hand position. There's a clear instruction to play the first note with the LH. It makes it slightly harder to phrase musically, but easier to not hit wrong notes! That's OK, at this level.