When traveling internationally, there is usually no need to take your passport on the road with you. Some countries may have a requirement to keep your passport with you so check with the Immigration official at the airport if you have any doubts. For the most part having your passport with you at all time is not a requirement for the majority of countries you will visit.
Once you check in at your hotel, you should either leave it with the front desk in the hotel main safe or you can leave it in the in-room safe. If neither are available, just leave it in your room, someplace out of plain view. This and other valuables are much safer in your room than on your person.
It is a good idea to make a copy of the first pages that has your picture and details. Just in case you lose it, you can show that to the local consulate for your country and that will help you to get a replacement travel document much quicker. Of course you should not store the passport and the copy in the same place.
Also, when you go out to see the sights, just take your driver's license or some form of ID other than your passport, a credit card, your bank card and some cash in local currency. This is really all you need. There is no need to take a bulky purse or wallet with you.