I've just been to Tingaki in Kos and would recommend that, it's got some nightlife, but not OTT, good beaches etc.
The good thing about Kos is that there is alot of scope for trips etc, 15 minutes by hydrofoil to Turkey an hour by boat to Nyssoros a volcanic island, which is also quite traditional too.
There are also trips to a number of other islands including Rhodes as well.
Somewhere else that I like alot, and similar in some ways is the North of Corfu, somewhere along the Northern coast Acharavi or Roda perhaps, not quite so much in the ways of trips etc, but still very good beaches and not too mad.
In Corfu avoid places like Kavos in the south ( club 18-30 land) and Sidari in the North ( fish chips & Mc D's), but there are many other places that will probably suit you. In Kos much the same can be said for kardamena, which is the main party zone but many other places are fine.
Whether to book independently or a package really comes down to cost and how late you are willing to book. I always tend to be a late booker and found myself a deal with Olympic holidays for £220 for a week in Kos, that was in mid June. Obviously if you can find a cheap flight it may be wiser to look for local private accommodation as opposed to a tour operator.