Executive summary

Bitcoin is working fine and you can use it as you were before, possibly with slightly more reorg risk for a while.

Continuing risks

Although I am giving a general "all clear" at this point because the risk of serious issues has dropped substantially, some turbulence is still possible due to BIP91/BIP148 and the rollout of SegWit:

 - If you are using a full-node wallet (eg. Bitcoin Core), you should require 30 confirmations for high-value, untrusted incoming transactions until SegWit locks in around Aug 10. Although seemingly quite unlikely at this point, it is possible that miners could fail to enforce BIP91/BIP148, which could cause very long reorgs (ie. confirmations could disappear, allowing double-spending). This seems unlikely enough that I probably wouldn't require extra confirmations for "normal" transactions, but if you absolutely cannot tolerate any particular transaction being reversed, then waiting for 30 confirmations would be most prudent.
 - If you are using a lightweight wallet, you should require 30 confirmations for all untrusted incoming transactions until SegWit locks in around Aug 10, and again for a couple of weeks after SegWit finally activates around Aug 24. Long reorgs or even persistent splits are possible if miners fail to properly enforce the rules of BIP91, BIP148, or SegWit, and lightweight wallets are particularly poor at handling such things because they will blindly accept invalid, probably-soon-to-be-orphaned blocks.
 - I don't recommend putting much BTC on hosted wallets / exchanges generally. Exactly how the above risks would be translated to these services will vary depending on how each service handles deposits, reorgs, and double-spends.