From what I recall (based on 9 years living in Northern California, 8 of them working in Concord) that there will be quite a bit of walking no matter which way you choose.
IIRC, the SF Civic Center Amtrak "station" is a bus top near the (underground) BART Station. And, at Concord, the BART tracks are elevated (with either an escalator, stairs or elevator), There will likely be more than a little walking at Concord BART, even if you'll be picked up by someone at the curb. (I don't recommend any taxi service in Concord--too many bad experiences with them.)
There's a direct Amtrak-to-BART connection at Richmond, in the East Bay (just north of Berkeley), as both are in the same facility. (They don't share tracks because BART's are wider than standard gauge.) But that still involves some walking to go from the Amtrak platform to BART's.
A better alternative to reach Concord would be to take Amtrak to Martinez (the next stop north from Richmond), and have someone meet you there & take you directly to your destination. Concord is only a short (10 minutes max) drive from the Martinez Amtrak depot.
Hope this helps!