Sorry for the long post...
Firstly, I am not a coach, Dr., etc. and am only writing from my own experiences. This isn't really advice, just what worked for me, and what didn't work for a friend. Congrats on the weight loss. I too lost 45 pounds (diet and elliptical because running was too hard for me at the higher weight) then picked up running which helped me drop more weight.
Many half marathon plans are 16 weeks long, hey that’s 4 months! And there are tons out there for beginners.
Here’s my path to the half marathon distance (without a training plan – Although all went well for me it’s not advisable to do without a plan. Pick a plan and stick to it. If you look at what I did, I did work up to it, it just was kind of rapid. Now I always use a training plan for anything up to or greater than a half marathon.)
I started running Feb. 2012, doing 4 mile loops near my home. That was def. work, but with the miles I was doing on the elliptical it felt pretty good. Anyway, one night about the 3rd week in, doing 3 to 4 runs a week I felt great and kept going to 6 miles and still felt great, but called it a night. Mid March I worked up to 8 miles and was loving it... Early April I hit 10 miles. That one was tougher, but was a great accomplishment for me. A few weeks later I did 11 miles, then at the end of April I did 13 just out there going, no race or anything. I kind of wish I had done it during a race, but I was still stoked; My time was even surprisingly better than I'd expected doing a 10 min/mi ish pace.
I then signed up for a 12k race because it was a few weeks later. The half marathon "training" run gave me real confidence for the 12k and I pushed to an average 9:15 min/mi pace (again I was stoked. I know many folks are faster and many slower but I run at what works for me and am not worried about others). After the 12k I signed up for a 10 miler for a few months later, the end of Sept. That went well, but it was much warmer so was a bit tougher but I kept at it having my long runs be 10 miles.
My first half marathon race was Dec 1 and it was great. I've done quite a few since then and a couple of Marathons (much tougher on the body for sure), but I put a lot of time between the two full Marathons and don't like having to lay off for recovery which for me half marathons don't force me to have so much down time.
I do think the 10k recommendation is a good one though because if you're digging 5k's but looking to push things a 10k is perfect. I actually prefer them, but they aren't as common. If there's a 10k I just sign up for it, I love them. I find in a 10k I actually push my pace and abilities without trying to hold back too much; really focusing on the negative split concept. When I go farther than that I really think I play it too conservatively because I don't want to bonk during a race. It's as much a head game as a physical challenge.
Summary: I took longer for an "official" half marathon race, but did the distance much sooner. If you know of a race that you REALLY want to do, just to experience it, and you don't feel like your body is being beat up doing 6 - 8 miles at a pace you feel good at, go for it. Just don't try to get caught up in the excitement or go out with a fast friend and over do it. Just enjoy the experience. If it's just a race for race sake, then wait. My first half marathon race always runs right in front of my house every year so once I started running it was "the" half for me to do. It was fun. It still is fun. Pick something you'll remember and go for it (if you feel good). Just check the weather, it's such a factor as you start to go longer and it's really tough to figure out how your body will respond to warmer temperatures.
Side note: Everyone is different. A co-worker picked up running and is highly competitive. He pushed his pace to go really fast and did a half marathon distance pretty fast in less than 2 months of starting running. It caused him GREAT pain and several months of non-running recovery. He went too long, too fast with no recovery between any of his runs. So whatever you do pick a plan and stick to it, including recovery, hydration, nutrition and sleep.
Good luck and have fun.