Training Fields are small neutrally-owned tutorial areas on the map where you can go to learn or hone basic combat skills.
You can use the Training Fields to earn minor experience bonuses and increase Weapon Proficiencies without the risk of capture. Early on this might be useful and if you're a capable player you can get better results because the experience your party gains is based on your performance during a specific training session. For example, you can train horse combat by going around a track which is lined with destructible targets. Hitting only some of the targets means only some party members will get a little bit of experience, but hitting all targets will net you a nice experience bonus for your whole party.
One on one Sparring is a fairly simple way to get experience for your squad, presumably fighting more and higher skilled members of your party gives bigger experience gains.
When using a training field, sparring will give the most exp. to melee infantry units, less to melee cavalry, and very little or none to ranged units. Sparring against melee units will give the best exp. gain to all melee units.
Ranged weapon practice gives good exp. to all ranged troops (but never melee units). Hitting every target during ranged weapon practice gives a major exp. boost, so hitting all 10 targets at 10 yards will give more exp. than hitting 9 targets at 20 yards, and so on.
Horseback training gives a good amount of exp. to all cavalry units - however, trying to hit one of the targets with a melee weapon is very hard, because the targets are very small and so it's hard to angle your lance or sword to hit them properly, so if you are skilled in archery or throwing, then that gives much better results, as the training gives exp. to all cavalry regardless of weapon choice, more depending on how many targets you hit.
A useful technique is to train your troops in "batches". The more troops of a certain type you have (e.g. rhodok spearmen), the more experience each one gets. For example, if you spar against 4 rhodok spearmen, and those are the only 4 you have, the summary screen will tell you that 4 rhodok spearmen gained 36 exp. However, if you spar against 4 rhodok spearmen, but have 36 of them in total, the summary screen tells you that 36 rhodok spearmen gained 241 exp. (each). Even if you don't train that particular unit, the same bonus applies to all troops. The more of a specific troop you have, the more each troop gets in exp. Therefore, when looking to train an army at a training field, upgrade your troops when they're all ready to upgrade, as they'll gain experience and upgrade faster when there's more. This same principle applies to the training skill, and to experience gained from actual battles, so unless you desperately need good troops, try and wait until every one of a troop type is ready to upgrade to get a stronger army faster.
With Fire & SwordEdit
In With Fire & Sword, this structure was changed into the Mercenary Camp and has the alternate function of buying and equipping regional mercenaries. Oddly enough, there are no tutorial sections for With Fire & Sword, unlike the two previous games which have a tutorial scenario on the home screen and training fields for in-game training that isn't a battle.