Looking at future of next 4 years.
In Senate, Warren, Franken and Bernie should take the charge.
Unleash the Biden on Trump/GOP for sure. He is the elder statesman. He has that ability to appeal across the spectrum and understands the meaning of big-tent.
Obama should concentrate on cementing his legacy, prepare the I told you so speeches, help in the redistricting fights and tutor young Democrats.
I do like the idea of Biden being a very public face for the Democrats in the absence of a more proper leader (president or presidential nominee).
Unless he decides to run in 2020 (I don't think this is super likely), his decision to abstain from the 2016 race lets him seem above the fray. Like, dude doesn't give a shit about being elected/re-elected and isn't just trying to save face for a disastrous presidential run. He just had a very successful vice presidency.
He's also just about the perfect counterbalance to Trump's strengths
now, although what the electorate will want in 2020 could be very different (think how as someone mentioned here before John Edwards seemed like the perfect candidate on paper for 2008 after 2004, but clearly the nation was moving away from the Bush-era politics in both parties). But if he's not a candidate that's no problem - he can help lead a solid victory in 2018 and step aside once we have a nominee.
Obama can make appearances and rally in the sun belt states, states like NV, CO and VA that Clinton held onto with relative ease and AZ, GA, NC and maybe TX - states that are potentially moving in our direction. Biden's appeal is more broadly in the rust belt, PA, OH, MI, WI.
2020 presidential primary should depend on what happens in 2018 and honestly I hope we just get like ten or so decent candidates and see what happens. I think it's a lot harder to factionalize in a negative way like in 2016 when there's several legitimate candidates versus one very obvious frontrunner and one possible anti-frontrunner candidate, as we saw with Clinton and Sanders.