Hi! Thanks for posting.
TL;DR Yes! Monitor speakers will be a huge step up. Never tried the PreSonus Eris range. Shop around a bit first to look at alternatives.
Going from built-in TV speakers to a dedicated system will be a huge improvement almost no matter what you get. Now the fact you are looking at monitor speakers in particular PreSonus's range tells me you are looking for something for mixing/mastering.
With that in mind, I've never tried the PreSonus Eris range so I can't comment on their quality. However, for a huge leap up (not in price), I would recommend KRK classic 5s or a pair of Focal Alpha 50s as a nicer starter to you audio production journey.
Shop around a bit. There are some really good guides out their for finding the correct studio monitors for you. Hope that helps!
Hey in case you haven't got anything yet, I'll throw in my few cents.
Mic wise, I would get a dedicated microphone. When buying a headset usually the money goes into the headphones and the mic is just an add-on. I have a mic that requires an actual audio interface to run but I'd get something like a blue yeti or a Audio Technica AT2020 (USB). If you want something clip on I'd look at the mod-mic, maybe a little pricey though.
Headphones wise, I can only really talk about what I use for gaming which is a pair of Beyerdynamic DT770s. They are closed back but are really nice all the same. A friend has a pair of DT990s which are open back so that may be more your thing.
Hope that helps.