Towered airports and some with FSSs will have an ATIS. In addition to weather information, it also includes the runways and approaches in use, as well as any NOTAMs regarding runways and taxiways.
At military bases, they also typically include runway braking effectiveness information.
Many, but not all, non-towered airports will have automated weather broadcasts as well. Any airport with an instrument approach is REQUIRED to have automated weather available. However, these may not always be over VHF frequencies. I've flown into a couple of airports that only had NDB approaches (at the time) and the weather was broadcast over the NDB (MF) frequency. Then there are others that broadcast their weather over the VOR frequency if there is a terminal or low VOR (likely not owned by the FAA) located at the field.
These are even fewer and far between than they were when I was actively flying 4-5 years ago. Unless they're part of another approach, quite a few stand alone NDBs and terminal VORs are going the way of the dodo as GPS approaches are added to these airports. As a result, they've been replaced with VHF ASOS/AWOS systems that aren't associated with a navaid.