With DD#1, I was having mild contractions a bit at night but I just assumed they were BH. I went to sleep, slept fine, and woke about 6:30 and they were full contractions only 3 min apart. I woke DH, took my shower, downed an instant breakfast and went to the hospital. I was admitted, etc., but I still had a long wait yet before the "pushing." (Got to hospital about 7:30 and by the time I was in a room it was about 8:15, but I wasn't allowed to start pushing until 4 p.m.) That was because DD was sunny side up and they were trying to turn her first, and I wasn't fully dilated yet. So they broke my water around 11 and then later turned her, and I went through really harsh contractions (even after a non-helpful epidural which did squat) for several more hours before pushing began.
This time, I'm a little torn on a few things and I will ask my OB next visit. I would like to labor longer at home than I did last time, but since 2nd babies generally come faster, I don't know if that's a good idea. (Hospital is 30 minutes without traffic and traffic is likely if it's during a weekday.) I also don't know if I want to bother with an epidural since it didn't seem to "take" and I was still in so much pain so why bother doing it? So if I don't need an epidural I can walk around longer and maybe things would have moved a bit faster, I don't know.
For you, I would follow what my OB says, because you don't know what's going to happen. I thought I'd have more "warning" than just waking up and it being time, but then I was still at the hospital a long time, so even the timing of your contractions can't really tell you how dilated you are or if it's time to go NOW. Also, your water might break, and I know for my OB that's an automatic come to L&D.