Do I need to change my network set up?

We have a basic internet set up.  100/100 plan and an old mr424 router.  Router is connected in the basement via coax and CAT.  Overall performance is OK although my daughter is in a 2nd story bedroom that is furthest away from router in basement and she can lose signal from time to time.  First floor seems fine and other bedrooms on 2nd floor seem OK as well.  Not much of a signal outside on the patio.

Wife is starting a new job and will be working from home (bedroom office on 2nd story) and will be connecting remotely to her companies servers daily.  I work from home from time to time as well.  So in all there are 3 laptops, 2 chromebooks, a few ipads, our phones, two TVs, a firestick, two cell phones and multiple Alexa devices (granted, not all at the same time obviously).  We also are just on the standard FIOS TV system, not the FIOS ONE system.

I have researched options like mesh systems, upgrading to a better Verizon router and maybe an extender, etc.  But I think it has left me more confused than ever.  I am also operating on some assumptions that may not be true.  For example I was curious why the mr424 router had both the coax AND Cat attached coming from the ONT, but I had read they both have to be connected.  I also read if you disconnect the CAT, the wifi will still work but then the TVs will drop their guides as they are reliant on the CAT connection to the router from the ONT.  This is also an issue if you switch to a third party router - you loose the TV guide on the TVs.  

Speed wise, I am working from a main floor office above the router in the basement and I am only getting download speeds of around 25-30 even when there aren't many devices active on the network.  Obviously the signal drops upstairs to even slower speeds.

Basically, I am looking for suggestions to strengthen the signal throughout the house and not be a patchwork solution.  I don't mind paying a bit of money but I also don't want to spend money unnecessarily.   I appreciate any guidance one can give.

PS - There is a coax connection in the bedroom where the office is, so is there a hard wire option to connect wife's computer?  Thanks again.