The answers to these questions are very dependent on the field of study. My perspective is mostly from economics and some statistics, so things might be different in other fields.
From an unknown school, it will be very difficult, but not impossible, to get into a really top program. Top programs tend to be risk-adverse, and you're facing pretty stiff competition in the applicant pool.
The biggest issue is that they are going to have a difficult time evaluating the difficulty of your coursework. Assuming you get that far, you're often going to be compared to a second-tier candidate from a top university. They know exactly how to evaluate the latter's grades and can even compare him/her to past students from that school. You will be much more of an unknown, which is not a good thing.
To make up for this, you're going to need exemplary letters. But that's the second issue, your access to great recommenders is more limited than someone at a top program. Letters of recommendation are often put in context by comparing them to previous letters from the same person, and here you might run into the same issue as evaluating your grades. Personal and professional relationships can also come into play, and your recommenders are less likely to be well-known in the field. Letters from people who aren't active researchers are much less valuable as well, and depending on the college you might not even have good access to active researchers.
At minimum, you'll need near-perfect grades in the most difficult courses offered. This probably means being one of the "superstar" students at your school, easily the best in your class. You'll also need fantastic letters from the top researchers that you have access to, and, for econ at least, significant research experience.
For your second question, general work experience is worth almost nothing in economics. All that matters is research experience. If the work you're doing isn't contributing to something that will eventually be published in a peer-reviewed journal, then it's extremely unlikely that it will be helpful.
I should also mention an option that is often available in economics, though I don't know if other fields have analogues. And that is to take a job as a research assistant for a professor or research group. These are often 2-3 year jobs that offer a chance to do real research with just a BA. Ideally, you could apply to graduate schools in two years with a coauthored paper on the way and a fantastic letter of rec from an active researcher.
Classic examples in econ are working for the Federal Reserve or for a professor's research group (example).
