You can automatically place a via while the Trace tool is engaged simply by tapping the number key that corresponds to the copper layer you want to transition to.
For example, if you're routing a trace on Top Copper (which is copper layer #1), and you wanted to route down into layer 3, simply hit the '3' key and a via will be placed to connect your trace from layer 1 to 3. Layer 3 will automatically become the active layer so you can continue routing on it.
The via will be placed at the last point you clicked in your trace path, not where the mouse cursor is.
To configure the geometry of the vias that are placed using this method, click the Via Size button from the Trace flyout.
From there you can select the via class you want to place during routing.