Welcome! Here's a brief tutorial. Click each image to maximize it.
Assuming you have enough resources and command points to purchase a land unit and transportation (resources are like money and command points control how many units you can have on the map; both are set by the scenario designer), you select the unit at the top of the screen and then its available transport in the bottom:

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Take a close look. The German Infantry '43 is selected. It alone costs 45 RP and 3 CP (resource and command points respectively). If you are looking at available transports, you see the the cost will go up to 75 for infantry and truck (45 + 30) and you will need 4 available command points (3 + 1) to place it on the map. As expected, the halftrack is more expensive in resources (105 = 45 + 60) but the same in command points. With the infantry unit still clicked, you click either the truck or the halftrack to add it to the purchase.
For artillery, that's where you may see the horse transport offered as well:

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In this picture, two infantry units have been placed, one with halftrack and one with truck (I clicked on one and hovered over the other to show both). Note the small halftrack and truck icons in the top right of their icons, showing that they have transport. Until they move, however, they are shown on foot:

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Here's another picture showing the same arrangement for two artillery units. Note their transport icons, too:

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This picture shows the infantry units after they have moved someplace. They are now in their transports, not on foot. Notice some differences in their ratings. The truck is a bit faster than the halftrack but the halftrack can defend itself better and actually has some offensive capability:

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Similarly, this picture shows the artillery units after they have moved and you can see the differences as well here:

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So it's all a matter of trade-offs and balances between cost versus maneuverability and combat strength.