i wouldn't worry to much about when you go into labour, its usually quite tame and uneventful and SLOW.... you might start to get period pain, and then it progresses.... there is LOADS of time to get to the hospital (usually, i know there are the exceptions of people's baby's coming in an hour etc, but you'd have to be pretty dam lucky/unlucky for that to happen)...
you could call an ambulance, or if you are going to the hospital early i don't see anything wrong with driving yourself, i know they would advise against it, but if its in very early labour with just mild period pain contractions, it could be done.... or call a taxi! ambulance or taxi - easy options...

its usually only in the movies where your waters break in some public place and thats the first inkling you have of labour, again i know there are exceptions to the rule...

i have a student midwife who has been going to my appointments with me and who will be present at my birth, maybe if you don't have a doula, you could ask if there is a student that would like to be present, and meet them before hand... chances are they would be dying to see a birth... my student has two kids, one c-section, and one VBAC so is a wealth of knowledge through experience... just an idea.

honestly the going into labour while exciting is not terrifying, as said above... its slow and steady in most cases.... i stayed at home for a good 12 hours before going to the hospital, then had another 15 hours at the hospital, hopefully yours will be quicker than that, but you can see there is loads of time to get to the hospital before the real action takes place (usually!)

good luck!