New After effects user

If you are new to After Effects the first thing. you should do is open the Learn workspace or the Learn section of the Home Screen that opens by default in After Effects. A few minutes here learning the basic UI and workflow will save you hours of frustration.

The second thing you should do is consult the User Guide. Links can be found on the After Effects product page and on the landing page of this forum. Before you start tackling projects like the examples you posted you have to start with some real basic workflows so you can figure how things work.

The first example you posted can be done with shape layers, the second is not much more than transform property animation. You won't find a specific tutorial that tells you exactly how to do either. but some time with the User Guide will help you get started. 

Here's one important thing that most new users miss. After Effects is not a video editing app. It is a motion graphics and animation app designed to create short sequences and shots that cannot be created in an NLE like Premiere Pro. If I were doing either one of your sample videos I would never try and do it in one comp. No comp would be longer than a phrase and the only time a comp would contain more than one shot is when the transitions could not be done in an NLE. The first example with the animated lines would be broken into 20 or more comps, each comp would be a separate idea, some would not even have a background so I could edit the overlays and transitions in Premiere Pro. You concentrate on one animation, one idea, that becomes a comp, then you work on the other idea in another comp. Then you combine the rendered footage for each comp into the final movie using Premiere Pro. The only time you would nest or combine several different shots (ideas) in a single comp would be if the shots were very simple, they required no manipulation for timing or audio, and they rendered fairly quickly. 

I hope this helps. Spending 3 or  4 hours watching and doing some of the basic tutorials, and watching the tutorials on rendering will save you days of frustration.