Hello Jeff and welcome.
I agree you have made the best choice in opting for a night on Rhodes rather than an awful early start and several hours of time-wasting in a place you didn't want to be!
Andy Ward, our resident travel expert has just posted the most recent ferry schedules which I think you will find useful. Unlike the official timetables published by the ferry companies, Andy's consolidated timetables show just the crossings between Rhodes and Symi (with some extending to Kos) and, most importantly, include all the available scheduled ferries and show their departure/arrival areas on Rhodes.
http://symiforum.com/viewtopic.php?f=8&t=2863It seems from Andy's list that you now have a choice of
three morning ferries, at 08.30, 09.00 and 09.30.
The first is the fast catamaran which will get you to Symi by 09.20 and that leaves from the Kolona harbour on Rhodes. The 09.00 ferry is the Sea Dreams 'Symi' which takes a much more leisurely two hours to make the crossing and at 09.30 you have the Panagia Skiadeni which includes a sightseeing stop at the Panormitis monastery so you won't be in Symi harbour until 13.00. Both of these boats leave from Akandia, the more distant port area that Kojak mentioned.
You have plenty of choice there! Personally, I would (in fact, do) always go for the 08.30 crossing but the choice is yours. Just remember to let your accommodation provider know which boat you will be on so they can arrange to meet you when you arrive on Symi.
I don't know which map you were referring to but this one from the Dodekanisos Seaways website shows the two harbour areas and you might find it helpful to consult it in combination with the location map for your chosen Rhodes hotel.
As Kojak mentioned, Kolona harbour is relatively central whereas Akandia is a little way out and unless your chosen hotel is very close by it's probably best to get a taxi. It will save your legs and you won't have the stress of having to find the port. If you tell the taxi driver which ferry you are travelling on he will take you right to the boarding ramp.
If you have more questions between now and September feel free to ask. We all love talking about Symi!