If the content on the old is only so-so, and the new one is better / more recent, then 301 the old to the new, matching up each old page to its corresponding new page as best you can. You'll get the benefit of the links that were pointing to the old pages, but now they'll be helping the new pages.
This will consolidate your link popularity, making it more likely the new site will rank well. Otherwise, if you develop them both in parallel, you're splitting your links between the two. At the end of the day, it comes down to whether you want one stronger site or two relatively weak ones. I know which way
I'd go with it, but in the final analysis it's up to you...
You definitely want a 301 if you want to pass link popularity. A 302 won't do that.
In my experience, most people not only don't pay attention to the URL, a good number of them don't even know what the address bar is or where to find it. What they pay attention to are the words/pictures on the page.
The only reason I can think of why 301-ing an expired domain to your new one would potentially cause a problem would be if the expired domain had been at some point used for nefarious purposes and gotten itself blacklisted for some reason. Google says (in theory, at least) they "reset" domain names when they change hands, but that doesn't seem to be consistently how things happen in Real Life.
It could potentially be a Bad Thing if a 301 redirect were to somehow associate your site with another having a naughty reputation or a dissolute past.
If there's any concern in that regard, you could try checking the Wayback Machine (
www.archive.org) to see what was on those domains back in the past. As long as they were clean, you should be fine with a 301.
My :02:
--Torka