First things first: when routing, set the grid to the finest resolution (0.01 mm in metric). There is no reason to use a coarser routing grid.
Next, use the OpenGL routing mode. It's better than the default canvas. In shove mode, it'll move other traces so that it never has a clearance issue. It won't move parts, but it will move vias.
Remember to click in the middle of a pad when starting a trace. With Preferences -> Magnetic Pads set to "when drawing tracks," the cursor should "snap" to the middle of pads automatically when the draw-trace tool is active. (Set Magnetic Tracks to the same setting, so you can be sure that when you continue a previously-drawn track that doesn't end on a pad, that it starts at the end of that track.)
And remember, as you draw your traces, clicking to set an "anchor" point is good, and you may have to use the [/] key to change the track "posture," which is how it wants to route from each "anchor" point.