I'm not sure that the case begins with George Papadopoulos. He was a minor staffer. It's unclear that he could testify against anyone else even if he did anything. He is indicted for lying to the FBI. This is the kind of problem that happens when law enforcement uses the following pattern:
LE: Do you realize that if you did [blah blah] that you could be prosecuted for ... So, did you do [blah blah]?
Lawyerless: No, not me!
And there's the crime. Lied to the FBI. Because the FBI only asked that question because they already had proof of [blah blah]. Almost universally good for an indictment, this can be hard to prosecute at trial. The problem being that [blah blah] often is not prosecutable. If it were, they'd just indict for that. So we get to the Bill Clinton problem. He lied, under oath, about something that really didn't matter (in Clinton's case, cheating on his wife).
The only real hope of getting to Donald Trump from Papadopoulos would be following a chain. But to date, Papadopoulos hasn't lead to anyone else.
More serious are Paul Manafort, who was a senior campaign official, and General Michael Flynn, who directly advised Donald Trump. There's also Rick Gates who worked for Manafort and was indicted at the same time.
The problem from a news perspective is that we don't even know what, if anything, those people are telling the investigation. Mueller has been close lipped. So anything we've heard beyond the indictments themselves is merely rumor.
Flynn is also charged with lying to the FBI. This seems to be based on conversations he had with foreign officials that were recorded as part of our routine surveillance. Apparently, he answered FBI questions about those conversations incorrectly. This seems to have been caused by answering questions without a lawyer. He has taken a plea deal. Some speculate that this was to protect his son from more serious charges. But again, rumor rather than confirmed facts.
The danger with Flynn (for Trump) would be that Flynn might testify that he performed certain actions at Trump's direction. As of yet, there is no evidence that Flynn has said anything like that.
Manafort and Gates
Manafort is indicted (PDF) for misrepresenting himself on lobbyist paperwork. He maintains his innocence. This is complicated by the fact that many other lobbyists interpret the rules the same way that he did. So if convicted, this is likely to lead to other convictions and/or changes in how lobbyists work and file their paperwork. Rick Gates is mostly charged with the same things, although there are apparently some things Manafort did alone or before the assistant was involved with the paperwork.
It's worth noting that the Manafort and Gates indictments are for activity prior to the presidential campaign. There's nothing linking that activity to Trump. The only link to Trump is Manafort (and Gates). There is a link to Hillary Clinton campaign manager, John Podesta. The lobbying group that he founded with his brother Tony worked for Manafort's company on the relevant issues. Tony Podesta has resigned from the Podesta Group since that came out.
There is a legal theory that Manafort might know something about Trump that he could trade to get out of his indictments. It's unclear what that might be.
There's a rumor that Manafort may be facing more indictments, presumably based on information found in the investigation. Still just a rumor at the moment. No confirmation from Mueller.
It's my current opinion that Mueller will eventually close the investigation without reaching Trump. That opinion could of course change if they indict for something related to Trump. As is, we have two people for lying to the FBI and two people for filling out paperwork incorrectly (albeit consistently with how the rest of their industry was filling out such paperwork). None of them have actually testified about illegal behavior by Trump as far as we (the public) know.