Utility is kind of a broad stroke, there is so much in the way of variation that “a good starting point” is hard to define unless the target is known. Utility can be, among others, aviation, maritime, military, drug runners, clandestine communications, etc, etc. What most utility stations have in common are relatively low power (compared to BC stations), short duration transmissions, and unpredictable transmission times.
The low power means a decent antenna, something with some capture area, is a must for happy Utility monitoring. The built in whip or a 15 foot piece of wire might work for broadcasters that often run 100 kW or more, but will leave you wanting and discouraged with utility stations that are typically 1500 Watts or less, often much less. In general here the answer is the biggest antenna you can find space for. If you have room for a 75 foot wire put that up, if you have room for a half wave 160 M dipole put that up (my single feedpoint full sized three band 160/80/60 M inverted V is one of my favorite “general” receive antennas). However, remember that a really long wire antenna (in reference to wavelength, not measured size) does exhibit directionality. So if you have enough room for more than 100 feet of antenna you might want to give some thought as to how you lay it out, maybe point the possible lobes in an interesting area of the World. Some antennas are broader banded than others, and some are lower noise. I find that an 80 M full wave sky loop is a good utility antenna being low noise and roughly omni up to a pretty high freq, if you have the area for such an antenna and are willing to build it.
Radios for utility are an interesting discussion, and one that could take pages just to scratch the surface of. In my opinion the SDR, with its graphic representation of the spectrum, is custom made for Utility work. Utility transmissions tend to be short duration and sporadic in nature, easy to miss as you tune around with a traditional radio. A waterfall display that allows you to see a transmission even when you are not tuned directly to it is a great help in finding new or unlisted utility signals. Naturally if you are tuning to known voice networks the SDR will be no more useful than a traditional radio, other than the adjustability of the filters. For digital and odd modes (other than voice or CW) the waterfall of the SDR can greatly speed identification of the mode, and from mode often the service. Considering that SDRs can be had from $40 up I would say an SDR is a good starting place for the utility listener who is technicaly savvy.
The fact that many of the “interesting” utility signals, such as military exercises and drug operations, will quite likely not be on known networks is something to keep in mind. This sort of goes to the where, and when, to listen. Sure, for example you can tune to the known HF-GCS network and hear some phone patches, EAMs, and occasional sit rep from aircraft, but the really cool communications tend to be on discrete frequencies, unpublished and unknown until reported on a utility forum/web site/chat. These kinds of things often change fairly quickly also, so that a given frequency might only have interesting traffic for a day or so and then gone. Utility listening is much more dynamic than most other types of listening.
As has been said, generally listen outside the broadcast bands. List the BC and ham bands and go everyplace except to them. But select the freq based on the time of local day and what you are trying to hear. Want to hear Asian maritime operations? For example, if you are in the US it would not do for you to be trying frequencies below 10 MHz at your local mid day. Look at a map and determine were the daytime and nighttime is, plan your frequencies around that.
The UDXF group has been mentioned already, there are other similar groups on the web. If you are familiar with IRC there are a couple of utility oriented real time IRC chat rooms, such as #wunclub on the StarChat network (and a Google search can show you how to get there even if you are not IRC active). The advantage of Chat rooms are that you can ask questions and see what other people are listening too in near real time (assuming anyone else is active at that time). Remember what I said above, utility stations are often short duration and transient.
T!