Write an android application What is the best way for me to go about writing a new Bitcoin wallet application? Is piggybacking off API's like blockchain.info or coinbase a good tactic? Would the application then be OK to distribute (are said services open source?). Are there alternatives? Can I run my own Bitcoin server and write my own API to go with it instead? What would the steps be involved in that? Would your average punter trust a (legitimate!) wallet that I create or would they feel safer using an app that is built upon a renowned service?
You can fork Bitcoin Wallet, an SPV implementation of the Bitcoin protocol:

https://github.com/schildbach/bitcoin-wallet

It's GPL licensed.