The things you win (that is, you don't have to pay for) are souvenirs. They'll be in your souvenir list. Souvenirs are arranged in the list by their pre-assigned number. Souvenirs that you haven't yet acquired don't show up in the list, but there's a blank spot in the list where they go. For example, if you have souvenir #1 and #5, the list will show #1, blank, blank, blank (these are where #2 through #4 go), then #5, then blanks through to the end of the list (the list goes up to #32).
Travel tickets are sometimes available in the shop in your Tamagotchi. If you don't see any now, check back in about 4-6 hours. You may not see any for a couple of days, but eventually they'll show up. You can use the time to get Gotchi points because you'll need them, since the tickets are expensive (several thousand Gotchi points each).