Question: What sorts of riding activities can you do with a horse that are non-competitive and recreational?
About the article "Equestrianism", section "Overview of equestrian activities".

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Horses are trained and ridden for practical working purposes such as in police work or for controlling herd animals on a ranch.
They are also used in competitive sports including, but not limited to, dressage, endurance riding, eventing, reining, show jumping, tent pegging, vaulting, polo, horse racing, driving, and rodeo.
(See additional equestrian sports listed later in this article for more examples.)
Some popular forms of competition are grouped together at horse shows, where horses perform in a wide variety of disciplines.
Horses (and other equids such as mules and donkeys) are used for non-competitive recreational riding such as fox hunting, trail riding or hacking.
There is public access to horse trails in almost every part of the world; many parks, ranches, and public stables offer both guided and independent riding.
Horses are also used for therapeutic purposes, both in specialized paraequestrian competition as well as non-competitive riding to improve human health and emotional development.
Horses are also driven in harness racing, at horse shows and in other types of exhibition, historical reenactment or ceremony, often pulling carriages.
In some parts of the world, they are still used for practical purposes such as farming.
Horses continue to be used in public service: in traditional ceremonies (parades, funerals), police and volunteer mounted patrols, and for mounted search and rescue.
Equestrianism (from Latin equester, equestr-, "horseman", "horse") more often known as riding, horseback riding (American English) or horse riding (British English) referring to the skill of riding, driving, steeplechasing or vaulting with horses.
This broad description includes the use of horses for practical working purposes, transportation, recreational activities, artistic or cultural exercises, and competitive sport. 
A crupper is used on riding saddles when riding a horse or mule with low withers in steep terrain.
Cruppers are most often used in activities such as endurance riding, mounted orienteering, competitive trail riding, and for recreational trail riding in mountains.
When riding, cruppers are particularly useful on animals with low withers, because this conformation allows the saddle to slip forward as the animal travels downhill.
Though not a substitute for a correctly fitted saddle, they are often used on mules, as many saddles, particularly those made for horses, are more likely to slide forward on a flatter-backed mule.
The crupper strap may be single or double (forked).
It usually attaches via a snap or buckle to a crupper ring at the center of the cantle of the saddle, or to a pair of rings (dees) on either side of the cantle.
Many riding saddles today are manufactured without properly placed rings, though rings can be added.
Or, in lieu of rings, a design used on English saddles utilizes a t-shaped brace that is inserted under the saddle between the rear panels and the tree.
Cruppers are not common on western saddles, but some English saddles are designed with crupper rings, particularly those designed to fit mules.
Cruppers are seen on the saddles of some horses used to work cattle with a lance (such as horses of the buttero in Europe), but not on horses used to work cattle with a lasso. 
Besides activities at church and school, there is a youth recreational park which has baseball fields and football fields for the brooks youth league.
Brooks’ road ways are often filled with bike riders.
Many bicycle races come through the area and often you can see bikers practicing for the races.
Many other activities in the area are just personal pleasures such as horse riding, riding ATVs, fishing, hunting, and other things.
Many people like to go up to a local gas station, called Huckaby's, to maybe get a pizza and hang out for a little while as well. 
Recreational use of the lake and its surrounding bushland reserve is severely limited, with prohibited recreational activities including swimming, water skiing, diving, mountain biking, horse riding, canoeing and kayaking, camping, and bushwalking.
Picnic facilities are available at four locations around the dam, with access prohibited outside of daylight hours.