We know the public won't accept Mueller getting fired so can't he just install someone who will severely hamper Mueller - refuse him access to subpoena anyone, refuse him ability to release any information etc.
But Robert Mueller isn't actually required to get permission to do those things. That's what the independent in independent counsel means. He is himself a prosecutor. He asks a judge for subpoena permission, not anyone in the Justice department.
Even if legally possible, refusing him the ability to release information seems unlikely to succeed. If he has actual information, he could always leak it as James Comey did, either directly or through an intermediary. Or write a letter to Congress and let them leak it.
The new Attorney General would have to pass congressional confirmation and there are enough anti-Trump Republican Senators (John McCain, Ben Sasse, Jeff Flake, Bob Corker) that a candidate would have to pledge to continue the investigation. There are forty-seven Democrats and two independents who would vote against almost any replacement, so it would only take two Republicans to join them to derail a nomination.
Until a new Attorney General is appointed, Rod Rosenstein would be the acting Attorney General. The never-Trump Republicans (and all non-Republicans) would prefer Rosenstein to someone who would fire Mueller.
Donald Trump can't fire Rosenstein directly. He would have to get Jeff Sessions to fire him. Or fire Sessions and then fire Rosenstein. That would make Associate Attorney General Jesse Panuccio the acting Attorney General. The next three are defined by executive order 13787:
(a) United States Attorney for the Eastern District of Virginia;
(b) United States Attorney for the Eastern District of North Carolina; and
(c) United States Attorney for the Northern District of Texas.
He might have to go down the list looking for someone to fire Mueller. Or even write a new list, although he's still stuck with the list of United States Attorneys.
We don't know what would happen next. Perhaps Trump continues his charmed existence and his supporters believe him over the media narrative. Perhaps this is just too much and he is impeached or resigns.
It's worth noting that Richard Nixon and Bill Clinton were both guilty of the things of which they were accused. Nixon was forced to resign. Clinton survived because he was guilty of something that wasn't criminal at base (cheating on his wife with an employee). Here, we still don't know if there is actually anything at the bottom of the investigation that relates to Trump himself.
We have Russians with no obvious relationship to Trump being accused of "election interference". We have Americans connected to Trump who are indicted for lying to the FBI about their own behavior and Americans indicted for things prior to the campaign. So far, no one has publicly associated Trump with a criminal act of their own knowledge. People have speculated about it, but the people with actual knowledge have not said so publicly.
Potentially, Trump could be impeached. But potentially it could have no immediate consequence. We won't actually know until it happens. Newt Gingrich was convinced that impeaching Clinton would lead to a Republican landslide in the Congressional elections of 1998. The actual result favored the Democrats slightly. We don't have enough examples of this kind of prosecution to say what would happen. It could go anywhere from nothing to Trump being executed for treason (after his impeachment).