Turing machine that applies homomorphism on input string
I could really use some help conceptualizing this problem and how to think about Turing machines in general.


Turing Machines are formal logical models that can express computation. You can think of them as mechanical machines that perform read and write operations on one or more tapes containing symbols. These operations are not performed randomly but based on a table (called the transition function) 
which is the true essence of the Turing Machine, without it the machine could not even move from the initial state. 
This table specifies what the machine must do in each state, based on the 'settings' of the "machine world state", for example:


If the machine is in state $q_1$ and it reads a '$1$' on the input tape, then write a '$0$' on the working tape and move left on the same tape.
If the machine is in state $q_3$ and it reads a '$0$' on the input tape, then go to state $F$ (HALT).


These are some basic description of a TM, if you want to know them better (and I believe it is mandatory to master them if you want to continue with this discipline). Then you need to open a textbook and dig into these concepts.
Now going back to your problem, you might find it useful to throw down a high-level description of an algorithm (turing machine) that performs the required transformations on the input string; then you can proceed from here, after you have learned how to formally define a turing machine, and draw your model.

Possible algorithm: 


start reading input tape
if input at position $i = 1$, then write a $1$ on the working tape at position $i$ and a $1$ at position $i+1$, then move right on the input tape.
if input at position $i = 0$, then write a $1$ on the working tape at position $i$ and a $0$ at position $i+1$, then move right on the input tape.


To this you need to add the implementation for adding '$01$' at the start of the string.