I was playing music on piano (I am fairly new), and the treble clef turned to a Bass clef. I have two and I am confused as to how I would read the notes. I already know that I have to play on the left side of middle c, but do not know which notes correspond to which keys on the new clef.
Its not the case that you always play left of middle C with the left hand and right of middle C with the right hand. As a general rule what you do is to play what is on the upper staff with your right hand and what is on the lower staff with your left hand.
Now for many pieces that will be pretty much as you say - right hand above middle C and left hand below - but not always. If the piece goes very low then the upper staff might convert to bass clef because its easier to read. Similarly the lower staff might convert to treble clef if the piece goes very high.
So what you need to do is to play the notes indicated on the upper staff with your right hand. If that has a bass clef then read it as a bass clef. If it has a treble clef then read it as a treble clef. The lower staff is for your left hand and you do exactly the same thing - play the notes indicated.
You may have found that playing notes on leger lines (the short lines above or below the staves) is tricky compared with reading the notes within the 5 lines/4 spaces.If a section of the piece needs to be lower (as this case) than the treble clef notes can be written within the 5 lines/4 spaces, There are two options. Use loads of leger lines;or, write the section in the bass clef, where at least, the dots are easy to translate into easy to play notes.
The same occurs in the opposite direction when all the notes need to be higher than, say, middle C for the left hand. Here, both of the clefs will appear as treble, so we're reading the l.h. using treble clef names. Sounds like it's complicating the issue, but it's actually making things a little clearer.
Explaining the above image:
In the first line, you might see these three measures and find it difficult to trace with your eyes. You can see right off the bat that there are some "middle" notes in the first measure, then some "low" notes in the second, and finally "high" notes in the third measure. But reading these fluently becomes a matter of how skilled one is in quickly counting more than a few ledger lines. After, say, three or four, it becomes somewhat laborious to count further. (Think of it: many children in school learn their multiplication table up to 9 times 9, but how many have mastered up to 20 or 30? Musicians would likely prefer not to have to read a large number of ledger lines, if possible!)
However, in the second line, you come across some clef changes in the music. That's not all that unusual: since the notes of the second measure are low enough in the right hand, they can stay in the top staff and be written very nicely in a bass clef without any ledger lines. The same goes for the left hand in the third measure: the notes are high enough in the left hand, so they can stay in the bottom staff and be written nicely in a treble clef. There's really no limitation as to how long the music may be written in another clef. It may be short and temporary, or it may be that the entire piece is written with two of the same clef!
In this third line, you come across what look like blank measures... does that mean that the right hand should not play anything in the second measure, and the left hand should not play anything in the third measure? Not exactly. There are actually no clef changes whatsoever, true, but what you should notice is that the notes in the right hand are still low enough in the second measure that the author may wish to direct you to the other staff for your right hand notes; the same is true for the left hand's high notes in the third measure.
I hope this helps!
P.S., Middle C is in a different position for each clef, as you know: it's the first ledger line below the treble staff and the first ledger line above the bass staff. That doesn't change, no matter which staff you are looking at.