there are plenty of places to travel with a young one in the region... we started travelling with our son since he was 5-months, not counting the long haul trips visiting family in Canada / US, we visited Bangkok, Ho Chi Minh City, Taipei, Macau and Singapore for short family trips.
where you choose would largely depend on what type of holiday you (the parents!) are after, since a one-year-old would be happy as long as they rest well and eat well, and have some space to run around. but i would suggest to go somewhere with direct flight, reasonable distant between the airport and the resort / hotel. our personal choice is to stick with the international hotels chains where you are guarantee that they can provide baby amenities from cribs to baby bath (some with cute little bathrobes too!) to bottle sterilizer so it's less for you to travel with, and they usually are equip with high chairs, kids menu in the restaurants and kids facilities on site.
also, how many people are travelling would also be a consideration: it is just for a family getaway for the 3 of you? or are you planning to travel with helper / family that you may have extra help? coz if you have help you may be able to do more. while you can always get baby sitters at the hotel, we weren't comfortable enough to do so yet, maybe when he's 2+ and can communicate better we can give it a try.
we always travelled just as a family of 3 and that's why we have picked places that we have both been before the baby, so we are familiar with what to do and what can be skip with a kid. our first trip when son was 5-month was to Bangkok, we picked it as it's a short flight, we stay in a familiar hotel which has nice club lounge, kid friendly restaurant and nice pool. it was our trial trip before the long haul flight a month after so we wanted to keep things simple and not to worry about exploring. we were able to enjoy time at the pool, nice relaxing walks and spent most of the time at the hotel and surrounding, we also took turns to the spa while the other half watched the baby... they have good international hospital in Bangkok and English is widely use in tourist area.
our most recent trip was to Singapore over Christmas. it's a reasonable flight time, very clean and in general a very family friendly city. there are good parks outdoor and also many indoor play gym which worked well when it was raining. the east side was also a nice stroll with the little one and there were many families in sentosa, we also took our little one who was 13-month then to the universal studio as well, he got to try 2 rides and really enjoyed it.
for our next trip we are thinking of a resort vacation, we've heard good things from our friends with club med but they suggest to go from age 2 onwards where the kids can enjoy more activities (and for us as well!). i think any nice beach resort would be fun for a one-year-old too, our son enjoyed playing with the sand when we were at santa monica last September so maybe that can be an ideal too...
have fun planning =)