Do small airfields provide ATIS? If not, what weather reporting service is available?
There is no minimum definition of a "terminal" in and of itself. The "destination terminal" of a flight is simply its planned endpoint. That's usually, but not always, a man-made complex designed to facilitate human flight, which we call an airport. However, you can file a VFR flight plan to fly out to a friend's fallow field in BFE, miles from any airfield listed on a sectional:



It's then your responsibility, as it always is in VFR, to ensure you can operate the plane safely at all times, including getting the plane on the ground and then back up in the air without causing personal injury, property damage, infringing private property rights, violating noise ordinances, etc.

Sectional diagrams use the term "airport" to refer to any place that is registered with the FAA to be used for the purpose of aircraft takeoffs and landings. These places may be more specifically called airstrips, airfields, air parks, aerodromes etc to indicate relative size, construction, ownership or facilities.

The only requirements for an airport to be documented by the FAA and appear on a sectional are:


The location must meet minimum guidelines specified in the Airport Certification Manual, including land grading for smoothness and drainage, obstruction/obstacle clearance in three dimensions, and minimum safety markings and wind/traffic pattern indicators that are necessary to support aircraft takeoffs/landings at the location (really the only thing every airport requires along these lines is a windsock; everything else is provided as needed for the specific location or as desired by the field's users). This usually requires a minimum amount of maintenance/upkeep by its owner for the purpose, even if it's just a grass field (it would have to be regularly mown and weeded to remove certain spiky or wood-stemmed plants, for instance).
The location must be registered with the FAA by its owner/manager for this purpose (they can register it as a reserved/private airfield, and then only limited information will be given on the sectional, primarily as an advisory that the location is available for emergency landings).


That's all there is to it. Besides a windsock which is the absolute minimum, no minimum facilities must exist; even the traffic pattern/landing direction indicators are optional if traffic can always use a left-hand pattern and determine the runway approach from the windsock. Rhome Meadows Airport (T76) between Fort Worth and Decatur, TX is a grass strip, with no lights, no tower, no fuel, no hangars, and the displaced threshold on the south end is marked with two white painted tires (no threshold markings to the north). Nearby Heritage Creek Airstrip is only slightly better, with low-intensity runway edge lights. Despite this, both are fairly active for unimproved strips, with about nine flights a day involving each one in some way (TO/L, T&amp;G).

Here's the airport section of the legend for FAA sectionals:



You can see that there's a wide array of symbology available to indicate at a glance what the facility offers; control tower or no, length and configuration of hard surface runways if any, usage type, presence of FBOs for fuel/maintenance, navigational aids, and radio information including ATIS, AWOS, ASOS, control towers, UNICOM etc. Obviously a control tower needs a radio frequency, but there is no requirement that a towered facility have ATIS. ATIS is usually found at towered airports as its purpose is to reduce controller workload, but nontowered airports can have ASOS/AWOS stations which provide practically the same information regarding weather, without the superfluous runway information (at nontowered airports, the pilots usually determine the approach/departure direction). If an airport has no reporting whatsoever, you would usually tune to the nearest listed AWOS/ASOS.