My 2 cents is Bush Sr takes over. Fires Haig shortly after, replaces with Kissinger who he knew from his time in China, if not earlier. Keeps most of Reagan's picks. Reagan tax-cuts are still on the table, but Bush sees that they'll produce and an unwieldy deficit. He has a degree in economics after all, and he did not believe in supply side economics. Still it would be in poor political taste to not continue Reagan's agenda. However keep in mind that a lot of people vote for Reagan just because he was not Carter, they might not be so hot on Reaganomics, giving Bush some cover to roll back some of Reagan's suggestions. It is likely he will focus on more on cutting regulation and red tape than taxes. Tax cuts are perhaps smaller in scope from the outset, with more tax breaks for oil and gas industry. He knew the oil industry inside out, and he probably knew the oil glut was coming. It is likely he will push for reforms which lower price of oil in early to mid 1981, much like the ones pursued by the Reagan administration in the same time frame OTL. Lower oil prices have the added benefit of sinking the Soviet economy deeper into trouble.
With Reagan dead the right wing of the republican party will have lost it's greatest champion. Bush was a moderate republican. He will pay lip service to the right-wing of the party, particularly in the aftermath of the assassination, but it is unlikely he will bend over for them. In fact i can see him successfully creating a coalition of moderates from both sides of the isle. The religious right will rail against Hollywood for it's sin and excess, once news gets out about why Reagan was assassinated, but i doubt the view points will gain mass traction in the mainstream. Let us not forget that the 1980 vote was as much against Carter as it was for Reagan, if not more. A lot of voters may not buy into the right-wing's claims about immorality in Hollywood. If any Hollywood celebrity, like Martin Scorsese, Jodie Foster, etc., are hurt, or killed, as retaliation for Reagan's killing, which isn't unlikely, then don't expect Bush Sr to back the right anymore. We tend to think of the political extremes back then to be more sane than their modern counterpart, but they weren't.
Domestically his term will be rather uneventful. He will probably not get the VP pick he wants until 1982, when congress will settle
Bush Sr will be more of a foreign policy president