Credit/debit cards. Two different kinds to handle places that don't take everything. Somebody at home to pay the bills when they arrive, or set it up for autopay. This will take care of most things.
Credit/debit cards that work in ATM machines. Check coverage (web sites will tell you where the machines are) for whatever network you use to make sure you'll be able to find outlets where you will be.
Cash. As much as you feel comfortable with.
Keep all of the above on your body 100% of the time, even when in the shower (yes, you can do that). Rather than spread it around, I keep all my eggs in one basket and watch that basket like a hawk. If you are a very careful, anal person who doesn't lose things, that should work well. If you do tend to lose things, however, then you might want to spread the stuff around with perhaps some emergency cash hidden in the handlebars or seat post.
Assume you're not going to get mugged. It's extremely unlikely and you cannot prepare much for it anyway.
Insurance for your stuff? Forget about it. It's too expensive. Just take reasonable precautions, which vary from place to place. You're common sense, assuming you have some, will be enough.