You can get off and on as you please without charge while in port. An important point to note is that the larger ships RCI, NCL, CC, OV Etc have to dock in deeper water and usually put on a shuttle service, some charge, some don't (we've paid up to £5 per person each way in some ports)
The smaller ships like Thomson and Island dock nearer to the centre of town in most ports as they need less space and can use shallower berths. In places like Barcelona, Livorno and Ajaccio you can walk straight into the centre without the need for a shuttle .
What a shame you want to start your cruising life with an informal cruise - part of the great joy of cruising is having the rare opportunity to get out the glad rags and get dressed up and feel a bit classy, rather than mix with the football shirt and tracksuit generation. I would also suggest a large modern ship with all mod cons rather than some of the converted car ferries that call them selves cruise ships that are out there. If you wish to research how to do it your self there is one excellent website that you are not allowed to mention here but if you look hard you may be able to spot it.
Not all people that go on informal cruises are of the football and tracksuit generation and not cruise boats are converted car ferries, most like us have been on formal cruises and now like to be more relaxed.
We are first time cruisers as well and set off in oct. We both like to dress up a bit for dinner too...however, i think it is wrong to judge others by what they might wear! Do you know how much footie shirts cost? They're not cheap you know. Judging by what you have posted, it seems that while some may look "classy" their inner self/attitude towards others is far from it!
Use the good Berlitz guides on this site to research cruise ships. The more informal are Ocean Village Cruises, Thompsons and Island Cruises. Then read about them on the cruise reviews on this site. I have just reviewed Ocean Village 1. There is a lot of information in the review and feedback has praised it. It is titled "Ocean Village - like it or loath it - you decide". I have cruised many lines over 12 years and think OV is a great introduction to cruising. As for the excursions I would reccomend the Olives and Ouzo itenery as Santorini, Mykonos and Rhodes are get off the ship and you are there (via tender for santorini) Kusadasi is an interesting port but please factor in a trip to Ephesus as it is wonderful (you can take a taxi but may not save overall against an excursion). At Limassol a taxi from the port should get you to sights (or shops)easily. No shuttles etc needed. You never have to pay to disembark a cruise ship. Have a lovely time whoever you cruise with.