A bank account with Internet access will suffice for most of your needs. Assure yourself that the security measures of your bank are top notch and only log into your account from a computer with high encryption (at this point in time, this means 128bit). Do not hesitate to update computers in Internet cafes to 128bit - even with a slow connection it should not take longer than 15 minutes and after a couple of times you can do this in any language. Remember to clear your cache and remove all files after your Internet session.
In any case, grant signing authority or power of attorney over your accounts to someone who will deal with a) the unforeseen and b) what is not possible through online banking. For example, some investments may require local supervision.
Finally, watch out for inactivity and reactivation rules for bank accounts. Some banks levy a hefty fee as early as six months after the last transaction.