I've learned that most exchanges and wallets support the Segwit chain now, but some do not (Exodus, for example). Will I run into any issues transferring BTC from, say, Coinbase (Segwit) to Exodus (non-Segwit) to Ledger (Segwit)? Will the middle link (Exodus which is non-Segwit) mess up the transaction in any way?
See here for Exodus wallet roadmap, scroll all they way to bottom.
http://support.exodus.io/article/96-exodus-wallet-roadmapExodus is not really a good BTC wallet, it can't generate a Segwit receive address to help the network fees, but you can't even set BTC fees lower than their default, any BTC movement out of Exodus will cost you a fortune compared to other wallets.