The biggest mistake new players often make in this game is ignoring a little button on the hero training speed. The arrow pointing upwards is there for promoting troops, essentially upgrading them. It is visible when you have some existing lower level troops selected. It is greyed out if you can't promote the troops as you don't yet have the required rewards. It helps you to make sure all your troops are high level and really useful in battle. When new players often ask how they can get rid of unwanted low level troops and want to kill their soldiers (wasting resources in the process), the correct answer is to just promote your soldiers. You can only promote the soldiers to the highest level available to you. You should Research the technologies required to promote troops from the New Home tech tree you need Machine Parts to promote / upgrade assaulters, Vehicle Parts to promote riders, Firearms Upgrading to upgrade shooters.

Training, promoting and healing troops only need the basic resources of Food and Wood. Researching the required technologies is in New Home category, so it is cheap and relatively fast to research these technologies. Promoting troops works exactly like troop training, so improving the speed of promoting troops is possible the same way:

• Refugees improving troop traning speed.
• Troop training speed bonus from Restaurant.
• Troop training speed bonus from the first set of S1 heroes. 
• Using Troop training or general speedups. You can get these speedups from various sources
• Researched technologies boosting Troop training Speed
• Spending diamonds

Events like the Holistic Growth phase of Alliance Duel and the Army Expansion phase of Full Prepardness usually give points for promotion. The rewards for promiting is usually the difference between the "old value" of the troop (at low level) and the new value of the troop at high level.