For most intents and purposes, it doesn't matter too much where you go for your undergraduate degree, so long as you have a good range of opportunities and apply yourself passionately to what you study. Now, that said, the alma mater you attend does matter somewhat as to where you might get accepted into grad school, which in turn can affect hiring committee decisions down the line when it comes to applying for tenure-track jobs.

The most important thing at the undergraduate stage is really to be passionate about what you're studying, and to apply yourself and seek out wider opportunities outside of your formal education (e.g. study-abroad programs, museum docencies, attending conferences, serving on student organization committees (say, a history club, for example). Those things will distinguish your applications into grad school and further down the line. But ideally, try to do them because you love them, not just to pad-out a resume' - sincerity is usually palpable.

For whatever it's worth, I attended a local community college for the first two years of my undergrad before going to a bigger public university. It was cheaper, and I got more personalized attention from some really top-notch instructors. Now I'm a post-doc at one of the world's top 10 research universities, and looking back, it feels like it was the right way to go.

To add another datapoint to what /r/reslumina says, I got my BA from a large public university (though an R1, "public Ivy" type), an MA from a similarly-prestigious private university, and now am finishing up a PhD from a top-5 program for my subfield. This is all in art history rather than history proper, but I think my experience roughly tracks.

Since I'm ABD I've mostly lost touch with the origins of the newer cohorts, but it seems like my American colleagues came from a mixture of about 40% top R1s (Yale, Dartmouth), 40% prestigious liberal arts colleges (Middlebury, Carleton), and 20% of a slightly lower tier of R1s (Temple, Penn State, University of Maryland). I can't think of anyone who went to a regional university or non-prestigious teaching college.

Of course, many of us also had MAs, and a number of my colleagues are Chinese, Taiwanese, and Korean, and I can't really speak to their undergrad experiences.

So, long story short, you can get in anywhere with a degree from anywhere, but there does seem to be a bit of an advantage to coming from a "name" school, if for no other reason than the fact that the professors writing you letters of recommendation are likely students or grad school peers of the people you'll be trying to study with in grad school. You have much more direct access to the network. That said, I am also a big supporter of state universities, so if you want/need to save money by going to State U, I say you should.