You should not be expecting any money beyond your normal pay and any non-discretionary expenses while traveling. If travel is so much of an inconvenience for you that you are expecting pay beyond your normal salary, perhaps you shouldn't be going on the trip. Most people I know look forward to going away for training & conferences.
If the expenses are significantly higher than your salary, the company should be issuing you a credit card in the company's name so that the money does not come out of your pocket directly. In this case, everywhere I've mentioned "reimbursement" below will really mean "the company will pay for via that credit card account."
In my experience, US employers typically will cover:
Your wages for any days you're not at home that you normally would have worked (so if you leave Monday & return home Friday, you get your week's pay; if you return home Sunday, you still get a normal week's pay).
Transportation (airfare, most often) to & from the location
Reasonable meals & snacks for the duration of your business (where "reasonable" means a meal you'd buy for yourself while staying on a budget, not going to extravagant four-star restaurants for each meal). Alcoholic beverages are usually not included.
Reasonable accommodations for the duration of your business. If your training is Monday through Friday, the company should be paying for a hotel Sunday night through Thursday night (assuming you have a Friday flight) or Friday night (if you have a Saturday flight). If you choose to stay in the location longer, you're responsible for your transportation, hotel and food on those extra days).
Reasonable ground transportation to get you from the airport to your accommodations, and from your accommodation to whatever location you're visiting for your business (training center, corporate office, etc.). "Reasonable" meaning public transit, a midrange rental car, or a taxi - not a limo service.
Typically (again, my experience is in the US), once you're outside normal business hours or otherwise not involved with the primary purpose of your trip, your time is yours to do with however you want, within reason (i.e., not getting into bar brawls or engaging in illegal/dangerous activity) unless there's an explicit expectation that you will be doing other work for the company in that time. Transportation, admission (tourist attactions, museums, events, shows, etc.) will be your responsibility to pay for because they're not what the company sent you for.
As for your email, simply ask what the expense reimbursement process & policies are. It should be spelled out in an employee manual or policy somewhere, but you've indicated that none exists (that you're aware of, anyway). Make sure you have an answer before you incur these expenses.
Save all receipts and submit them in an organized fashion - don't just hand someone an envelope stuffed full of slips of paper. Itemize them, compile them into a spreadsheet grouped by category (meals, transportation, etc.) & submit for reimbursement. Submit photocopies - not the original receipts (keep the originals in case they're needed later). You might even want to write a note on the receipt itself if it doesn't have enough information to clearly identify what it is.