I second what ksandness suggests, and his opinion of Osaka. The balance between cramming things in and feeling too rushed is different for everyone, but one easy way to solve it is stay in places where lots of attractions are close together and/or accessible.

Is there a particular reason to avoid Tokyo? There is so much to see and do there, and if you are comfortable navigating a foreign city's subway/train lines with English signage and helpful staff, most things in Tokyo are easy to get to. I haven't been to Hakone, but it is definitely on my list and have heard great things.

So I also suggest Tokyo and Kyoto with a side trip (Hakone, Nikko, Nara all good choices). It is the "standard" itinerary for a first-time trip, but there is a reason why it is suggested so often.