I'm trying to write a Turing machine that can convert a number in binary representation to a number in decimal representation. For example if the input on the tape is 1101 the output should be be 13. Where should I start from?
Since you don't have any questions in your post, it's hard to give any concrete tips.
Try to do the simpler binary to unary converter instead. And should also assume that you have as many working tapes as you need (say, one input tape, one counting tape and one work tape) and a separate output tape.
You should also pick that of LSB/MSB that suits you best (e.g. LSB).
The biggest task is probably computing $2^i$, so you should start making one machine that takes as input a number $n$ in unary, and outputs (on a separate tape) $2^n$.