Okay, after doing a bit of searching I managed to find enough info to make better searches with.
It's hard to search for something when you do not know the appropriate terms.

It looks like what has been used so far is PEX with crimp connectors. It looks like I should easily be able to pick up some tubing and compression fittings for the things I need to complete this one section for the bathroom. If I find a crimper inexpensive enough I may go that route rather than the compression fittings.

Are there common issues I should watch out for such as common mistakes or things to keep in mind that might not occur to someone when first working with this type of tubing?

The hot and cold lines come from one side of the basement and feed the toilet, sink and shower with the cold line also running to the outside faucet at the front of the house then the cold and hot lines continue on to the other side of the basement to feed up to the kitchen sink. It makes for a complex mess of pipes in that small area with lots of connections.

I would like to separate the hot/cold lines going to the kitchen eventually but figure for now I will tackle the bathroom line and then mate it up with the copper until I can put in a better setup possibly with a manifold and then disconnect the kitchen from the bathroom line and run it directly to the manifold.

Is there a preferable configuration for the connection of these pipes? I have cold water running in 5 directions and hot running to three.
I want to make sure that I do not cause any undue restriction and that I use the correct size of tube for each application.

Any need for tube to allow for trapping/releasing air? Code regarding placement of the lines or where I would have to add in shutoff or drain valves?

I can draw up a diagram of what I am looking at for the existing copper if necessary. It really is a mess. The kitchen hot and cold lines route halfway around the house to get to the kitchen when they could have gone about 6 feet from where the hot/cold lines are by the water heater. The former owner was a real hack.

Thanks.