The pro of segwit is: It makes transactions smaller.
While you still have to pay the same feerate (x sat/B), you will (overall) pay less for your transaction (about 20-30% for P2SH-P2WPKH addresses [starting with 3...] - and even more for bech32 [starting with bc1..].
You can basically think of segwit as the 'removal of redundant information'. This makes transactions smaller -> Fees lower.
Unfortunately bech32 aren't accepted for withdrawals from most of the exchanges yet. But you still can always recieve/send your funds!
I would suggest electrum (https://electrum.org/#download) as a light weight wallet with some quite impressive features. Electrum supports both types of segwit addresses.
For 'maximum compability' you should use the nested P2SH addresses (starting with 3.. ). If you are more into saving fees / not really dependent on exchange withdrawals: bech32 would be the way to go!
For hardware wallets: Ledger/Trezor support segwit. Not sure about KeepKey.