While I use Linux full time, it's certainly not a magical solution to a slow PC, the PC is extremely old and even with a lightweight Linux OS it will still be slow. The OS is only part of the problem, even if you somehow install a magical fast distro, your web browser will still struggle to run well on the weak PC.
There are easy to learn and well supported distros out there (Ubuntu is pretty good as it's so widespread meaning there's tonnes of support online) but no matter what you go for it will take some getting used to.
If I'm totally honest, if you have a brand new MacBook Pro I can't think of any reason to keep using that Dell, you may be attached to it but it's really out of date and several times less powerful than the mac will be. The mac will be so fast, I imagine the Dell will feel even slower once you get used to the mac. If you want more of a desktop experience there is nothing to stop you connecting the MacBook to your monitor, keyboard and mouse and essentially using it like a desktop. By all means don't trash it and keep it around for tinkering with, but it seems silly to use it when you have a significantly better machine.