IF you are going to skip the ship excursions, then you need to do your homework along with a good sense of adventure. A bit more than what Carl's suggests. That is about my last option. First, your local library can provide guidebooks and CDs. Travel shows (CDs) are great for overviews. Use the internet to id resources. Some guide books will specifically recommend guides, local tours, etc. Contact them. Plan your day in that city. Understand transportation options. Second, the ship will have orientation session for that city the day before and, generally, will have maps of where you are docking and transportation options to the city center.
Once you dock you will have a number of options depending on what you have learned about the port. In some cases it makes sense to head to the nearest TI and book local tours or meet tour guides, etc. Often the hop on/off tour buses are a good deal for a nice overview of the city. Another option is to walk the taxi line, looking for a good English speaking driver. A lot of drivers will be very happy to give a half or full tour for 50 to 100E. We generally are a family of four so it is far more economical for us to hire a driver for the day or part of the day. Sometimes you will find someone offering set tours at the dock. Use your judgment. We have never used a dock tour group because they were not are part of our plan. And on a few occasions we have found it is more convenient to take what the ship offers.
You really need to understand where your ports are. For example, the Rome port is about hour away via train. Athens is some distance from the port as is Ephesus and Pisa. The reason ship excursions can be expensive is that they are doing all the planing and you may have substantial travel distances involved.
Good luck, we have done six cruises and have been in all of those ports at least once and a couple three times.