I hear the underlying idea fairly often, the idea that "Pence might be worse!" if Trump were impeached or resigned for some other reason, but this is unlikely. If the president resigns or is impeached, it's a reflection on the entire administration and the vice president has virtually no sway. In the case of impeachment, that would mean a majority of congress believes that the president cannot serve, and if a majority of congress are swaying against the president, the vice president will have little pull over congress.
Beyond the simple result of impeachment being a broad rejection of the administration, Trump being impeached would also pre-empt dozens of his Administration staffers resigning or leaving their posts prior. Kushner, Bannon, and every other appointed Trump ally in his administration would be out. Priebus would likely have to resign though Priebus is more of the Pence-mold than a Trump acolyte, but he would have to resign because if you're Chief of Staff for a president who has been impeached or resigned, you can't possibly keep your job.
You can really only look throughout history at the results of impeachment or resignation. Every presidential censure, impeachment, or resignation is unique, so there's no pattern that can be gleaned, but prior to Nixon's resignation, there was a rotating door of appointees, chiefs of staff, influencers, and policy makers in his administration. His resignation was preceded by dozens of resignations (forced or voluntary) prior, conspicuously of course was the resignation of his vice president Spiro Agnew just months earlier. Gerald Ford was selected and confirmed as Vice President (after Agnew's resignation) only because he was respected, well liked in congress, was bipartisan, and largely without controversy, or strong, long-lasting political ambition. Prior to being selected as Vice President, Ford had no relationship with Nixon. He was from a different political camp as Nixon, was not a Nixonian Republican, had always supported Nixon's Republican opponents over Nixon, and acted largely independent from the President while he was House Minority Leader. Ford had no political clout following Nixon's resignation and his confirmation as president, he wasn't able to influence policy or foreign affairs in any way that was his own, he had to mostly follow the lead of congress (a congress that had started impeachment proceedings against his predecessor). Ford's only consequential action, of course, was pardoning Nixon. something that only the president can do and there is no recourse from congress on this (and ultimately, something that I think Ford was right in doing). Should impeachment proceedings be taken up against Trump or should Trump resign, there is a good chance that Pence might already be out as Vice President, much like Agnew had resigned (though, Agnew resigned because of his own political corruption independent from Nixon... Neither man even liked the other), though there isn't a Gerald Ford in the Republican party right now -- e.g., a prominent congressional Republican who is well known, well liked, or who has a track record of bipartisan legislation. Jim Boehner or Paul Ryan might have been that before they became Speakers of the House, but not anymore, and obviously Boehner is out of politics though he might have been the closest prominent Republican to a Gerald Ford.
This all said, I'd caution against the zeal in taking up every story as the smoking gun of impeachment, which many have done. I'm progressive and I'm mostly friends with just liberals or at least Anti-Trump moderates or Republicans (Largely... I live in a liberal state which has already had city councils call for Trump's impeachment), and the willingness to think This new randoms story is the smoking gun! probably hurts the chances that Trump would actually be impeached. Impeachment is a process and it's not easy, and whenever a story comes up in the Washington Post or New York Times about the latest Trumpian debacle, too many people jump to the "This is the smoking gun!" reaction, and of course, the news takes it's course, Trump is obviously not instantaneously impeached as a growing number think that's how it works, and then they're disenchanted with the political process which may end up help Trump or his supporters. If impeachment comes, it won't come swiftly, it will come with a groundswell of support for Trump opposition largely from Democrats, but also in places where Republicans in congress have shaky seats. Stalwart, consistent Republican strongholds are not going to suddenly embrace impeachment because there is no political incentive for them to do so, and considerable political risk.
Pursuing Trump's impeachment on whatever the latest story is isn't bad thing, but if you delude yourself into thinking that impeachment is imminent, whenever the latest negative, embarrassing, revealing story comes up in the Press, is a bad thing. Impeachment is a process and, personally, I think that Trump's resignation is a higher likelihood than Trump being impeached, but he'd do it in typical Trumpian style.
