Let's contemplate how any potential impeachment proceedings of President Trump could play out. We are going to assume a couple of things. First, that an impeachable offense will present itself but not one so bad that it would be a clear slam dunk. Next, we are going to assume that each member of Congress will act in the interests of their party rather than the country (not a big stretch) and treat this as a big Game of Thrones.
Let's review how an impeachment works. Under Article II of the Constitution, impeachment proceedings originate in the House and a simple majority is needed to approve. If that happens, the proceedings go to a "trial" in the Senate where a 2/3 majority is needed to convict and remove from office. So, unless the make-up of the Senate radically changes in the 2018 mid-terms, bipartisan support will be needed for an impeachment and subsequent conviction.
From a Republican point of view, on the surface, it would seem unlikely that they would support any proceedings. Let's go a little deeper, though. If they don't trust Trump, either for the perceived damage to their brand or because he strays too far from their conservative orthodoxy, then they could get behind impeachment proceedings. They control congress so they could certainly take it at least to a trial in the Senate. Why would they want to do it? We can think of a few reasons. If they view Trump as particularly vulnerable in 2020, Pence may, in their opinion, make a more formidable candidate. Trump could prove too unpredictable where, again, Pence may be much more likely to implement a conservative agenda. Or, they may get sick and tired of being put in the awkward position of defending his behavior and statements knowing that it could have a negative impact on both the mid-terms and the next Presidential cycle.
Remember, though, it takes a 2/3 majority of the Senate to convict. The Democrats could form an alliance with Trump to prevent a conviction. Why would they want to do that? Again, there are a few reasons. They may consider Trump a much more vulnerable candidate in 2020. They may hope to win the House in 2018 in which case the new Speaker of the House would be next in line for the Presidency after the Vice President. If Pence could be implicated in whatever impeachment charges are brought forth, then there is a chance for them to take back the White House. This is a far fetched scenario for a few reasons. First, it is unlikely the Democrats will take back the House. Even if they were to win the overall vote, with the amount of gerrymandering in the districts, it is almost an impossible task. Next, a Republican Senate would be hard pressed to convict both Trump and Pence with a Democrat next in line. For this scenario to work, you would need indisputable evidence of Treason on both Trump and Pence's part.
The Democrats would, however, have a vested interest in either running against a vulnerable Trump in 2020 or delaying impeachment proceedings until right before the election to maximize their chances for winning in 2020.
These are just a few scenarios and it is impossible to predict what charges could be brought and how it would play out. We are, however, fairly confident that politics will play a big part.