PE - generally, the larger and more established funds don't hire straight from UG. They prefer to not have to train, so they pick off top candidates that have training from the best IBs, and increasingly the best MCs. The smaller and less established funds might be a little more receptive to hiring non-traditionally, e.g. straight from UG or from due diligence service providers (BigLaw, Big4 accounting.) Exceptions, as you note, that come to mind are Blackstone, Bridgewater, Audax and some others that do hire straight from top UG schools.
HF - this depends on the type of strategy. The fundamental oriented HFs will hire similar to the way PE does. The more quantitative oriented may hire from UG (I don't know as much about these) but I think they like to hire more from graduate schools, e.g. Ph Ds in physics, mathematics, engineering, etc.
VC - these funds are more top-heavy, i.e. less junior investment professionals as compared to senior ones, than HFs and PE. Admittedly, I know little about these funds but I don't believe there are many inexperienced hires to investment related positions. If they did, I would presume they would want either technology and/or medically oriented (graduate) degrees.
My advice is to get the best possible training at an IB or M/B/B if possible, then think about jumping to PE/HF or VC.
Re networking, what's worked for me is to first solicit general advice and to just learn by asking questions and being very interested in what they are doing.