The right hand plays the upper staff, and the left hand plays the lower staff, regardless which clefs are displayed. The clefs only tell you how to interpret the lines and spaces; they don't tell you which hand to use.

When playing duets, for example, it's very common that one part will have both hands indicated in treble clef and the other both hands in the bass clef.

Edit: As Andrew Chin mentioned, sometimes there will be an explicit instruction as to which hand to use. Otherwise, the above holds.

An example of explicit instructions for which hand plays which notes, see What does the L-shaped symbol attached to C5 and G4 on the top staff mean?

Remembering where each note is is virtually impossible with 88 notes to choose from, looking at the music involved!

However, you don't need to remember each and every one, as there's a pattern that repeats several times up and down the piano keyboard, and the two staves in the music written for it only encompasses around half of those! Good news so far.

Every note on the piano has its own special place, and it works best alphabetically. Your do-re-mi may be more familiar to you, but it probably will slow you down here. Firstly, there are two do-re-mi systems, fixed do and movable do. Fixed do may help, as that means each and every C note is the white key to the left of the pair of black ones. At least that will give you a start to navigate your way round.

If, quite possibly, you are familiar with movable do, it won't help much: that one means the root of every key, though a different note, is called do. Quite confusing for a beginner.

So, let's hope it's fixed do (or C). Travelling up the keyboard, it's sequential, C, D, E, F (left of the 3 black keys), G, A, B, and back to left of the next black pair, another C. Using fixed do, the same idea exists, but substitutes tonic sol-fa names for the letters. Just hope you're not using the German system which incorporates another letter - H.

That's all the piano part, nearly. Using letters only, a good way to familiarise with where each lives is to make up words, and play those words. As in DAD, BADGE, CABBAGE, DEAF, BEEF, and so on, until you can look at a word and play it straight off. Black keys can be ignored for now - they don't play this game. For now, it doesn't really matter which octave you play each letter in, it's fun to mix 'n' match. But familiarisation of the instrument is an important, and in some ways detached part of learning music - where each note lives.

As far as the dots on the sheet are concerned, there is correlation. With 2 staves of 5 lines, it helps to consider that middle C is in fact in the middle. It has its own little line - no permanent one. When needed, it has a 'leger line' under the treble, or above the bass clef.

From that datum point, it's not too difficult to extrapolate where on the staves the other notes live. The oft-used mnemonic upwards for the dots on the treble clef lines EGBDF is 'Every Good Boy Deserves (something beginning with F), the spaces revealing FACE. In the bass clef, spaces reveal All Cows Eat Grass, unsurprisingly. Of course, it will be more memorable for your own silly sentence to be utilised.

I hope all that's a good starter in response to what I think the question asks. Take it all slowly, and get a good teacher!