The best procedure to follow with a new computer is to install the game itself (of course) and all of your Expansion and Stuff Packs first, then test to make sure new games behave as well as they should.
Then begin to build up a new user game folder by piece by piece bringing over your installed sims3pack content (DCCache), package based content and mods if any (Mods Framework, then Mods\Packages), installed worlds, Library files (saved lots and households) and SavedSims if any. And again test on the new folder to make sure everything is working right. Then, finally, try one or more of your older saves in the new environment.
Unless you are trying to streamline things by not installing or carrying over all of the content that you had before, as long as we are talking about the same "kind" of install (e.g., Origin or disc vs. Steam), even the Store and Exchange content stored in DCCache in those dbc and ebc files should be able to work upon being copied over. There are a couple of gotcha's to watch out for, like you will also need your ccmerged file from the older install's DCBackup folder or get one from another source, or else your store premium content won't work. And if you happen to be on a disc or Steam install (Patch 1.67), you will need your store content patch files because the Launcher no longer offers those up for download -- Patch 1.69/Origin players don't need to worry about this one.
All of this isn't to say that prior saved games won't work if you just install the required EPs and installed worlds. Sometimes they will. But there might be so much in the way of content substitution that needs to happen in-game that your sims, their lots, and the worlds around them may look entirely different than the way you left them all on your older computer. And although the sims themselves might not care too much, I don't think I ever want to see my majestic CC statues on overly ornate lots being replaced by tiny ficus trees or even worse toilet roll holders again, the novelty of that kind of thing wears off pretty quickly. :)