1. Are there better alternatives to Coinbase in terms of purchase fees and transaction?
2. Wouldnt it maybe be cheaper to buy LTC, move it to the exchange and buy BTC with it?
Great initiative that you don't go for the first exchange you see but look around first.
We have developed a tool for people like you, who are thinking of what exchange that suits them best. It's available here: https://www.cryptowisser.com/
In that tool (the Exchange Finder) you answer 6 questions, and you are then matched with all exchanges that matches your input. In that matching, you will also see relevant fees etc.
Good luck in your search!
All the best,
Cryptowisser