I am going to run two connections from an existing electrical outlet, one to another outlet, the other to a light switch. I was going to pigtail the wires in order to make the connection, however should I also put the pigtail connection in a box or is it safe with just the wire nuts and electrical tape?

By law all connections have to be in a box and the box must be accessible. It can not be covered up with any wall finish products.

Chuck

There re box fill issues that you could run into based on the number of wires entering a box and the size of a box.

By pigtail, I assume you mean connect all the whites together with a wirenut and all the blacks together with a wirenut and leave a small length of each to attach to the outlet. Electrical tape isn't needed and should not be used.

And of course, deal with the grounds.