I ran a 10 miler while 14 or so weeks pregnant with my first and am planning to run a half marathon in October. I will be about 4 months along then.
You don't have to run slow, but you should listen to your body. You can push yourself a little, but not too hard. I'm no longer in the mentality of wanting a PR at the October HM, but just looking to complete it and feel good afterwards. So I've cut my speed training out, but have continued with hill training as the October HM is a very hilly one. I do allow myself to walk when I need to.
Keep up with your training schedule as much as possible through the first tri, but understand there are gonna be days that you are just too tired, too sick, or otherwise just not feeling the run. And give yourself permission to take extra rest days and not overdo it. When you get into the second tri you will feel sooo much better, and you can be a little more rigorous with your training schedule, still understanding that you may need an extra rest day or two over what you were used to. Also don't be afraid to abandon a run in the middle if you just feel awful. I cannot tell you how many times I did that when pregnant with my first. I'd get 1/2 way through, feel awful and walk straight back home. The next time I ran, I'd feel fine.
Get used to taking a sport drink with you while running anything more than a mile or two. Water just doesn't cut it for me, I have to do something with electrolytes in it. Also, start eating your favorite sports gel/chews a few minutes before your run, and faithfully eat them every 30 minutes. And don't be surprised if you have to pee a mile into your run, or every mile of your run. Also don't be surprised if you start hitting a wall 2-3 miles in. Allow yourself to walk until you feel better, and you will get past the wall.
Run when you feel best during the day. For me, I feel my best first thing in the morning, so I get up very early (5am right now), eat a Gu gel, drink a glass of water and hit the pavement. I do a 1.5 mile loop so I can stop by my house and drink some water, use the restroom if needed, and do a second 1.5 mile loop. On the weekends when I do long runs, I choose places that I know have restrooms available and usually plan my runs in 3-4 mile sections so I'm coming back to my car/stopping by the restroom a few times. Then as soon as I'm done with my run, I eat something with lots of protein, like a Greek Yogurt or a protein shake, and get ready for the day.
Also make sure you're not running when its super hot out. You don't want to get overheated while pregnant - same reasons you don't want to go into a hot tub or sauna while pregnant. Raising your core temp too high for too long can be harmful to your baby. That doesn't mean you can't sweat while running. Just that you gotta listen to your body and have some kind of idea of when you're getting too hot. For me, I know when my head is throbbing with my heartbeat that I'm too hot.
You need to make sure you eat enough when you're running while pregnant. Its helpful to log your food and exercise in MyFitnessPal.com for a couple weeks to make sure you're consuming enough calories.
That's all I can think of now. If you have any questions feel free to ask!
Re: Training for a 10 k in November