Executive summary
Bitcoin is working fine and you can use it as you were before, possibly with slightly more reorg risk for a while.
The BIP91/BIP148 split didn't occur
As had been expected for the last week or so, the big potential split which caused all of the Aug 1 hubbub didn't happen, due to BIP91/BIP148 succeeding. There's a small chance that this split could still happen sometime between now and around Aug 10, but it looks very unlikely at this point.
Bcash split, creating an altcoin
If you had some number of BTC as of approximately 14:00 UTC, and if these bitcoins were secured by private keys controlled by you or in a Bcash-supporting online wallet, it is now possible for you to claim the same number of Bcash coins (BCH/BCC). Bcash is an altcoin based on Bitcoin; since it split, it is now 100% independent from Bitcoin other than the carried-over balances.
Bcash claims to have replay protection, but this is only a Bcash miner policy. Any Bcash miner could still replay transactions. Therefore, you can't safely rely on it.
Instructions for safely splitting your BCH from your BTC
Here are general split instructions. If your wallet software gives different instructions, use those instead.
These are the best instructions I could come up with quickly. Maybe someone will eventually come up with a more convenient way of doing it. Note that while I believe these instructions to be safe and reasonable, you are solely responsible for your safety here.
1. Decide whether you want to bother splitting your coins at all. Is it actually worthwhile to remove your coins from cold storage (or whatever)? If you take no particular action and continue using Bitcoin as normal, then your BCH is likely (but not guaranteed) to remain associated with the current private keys due to BCH's pseudo-replay-protection; you could therefore maybe claim your BCH later on if you want. If you want to claim your BCH now, continue to the next steps.
2. Create a totally new logical wallet with a new mnemonic, etc. Just creating new addresses is not sufficient: the new wallet needs to be completely separate from the old one. The new wallet's private keys should be under your personal control just in case your transaction is replayed or something else goes wrong. Make sure you have backups of the new wallet.
3. Send all of the BTC in your old wallet to your new wallet, and wait until this transaction has 30 confirmations. (If you are really itching to get your BCH, it would not be totally crazy to wait for as few as 3 confirmations, but there is some risk with that.) Note that sending all of your BTC in a single transaction is likely to be harmful to your privacy; if you care about this, you might want to send it in several smaller transactions to several different addresses on the new wallet.
4. On a separate computer that you wouldn't too much mind being compromised, install a BCash wallet such as one of the ones listed on bitcoincash.org. (Warning: some Bcash wallets are known to overwrite important files belonging to their Bitcoin counterparts.) Or use a Web-based exchange/wallet supporting Bcash. Import the private keys from your old, now-empty Bitcoin wallet into this Bcash wallet. Now you have BCH which you can handle as you wish. Note that BCH transactions may be very slow to confirm for a couple of days.
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.
If you can use private keys with no btc on it (because you just sent it to another wallet) will it work with ANY private keys? What if I have very old private keys that had some btc earlier and I'll import it in BCH wallet?