You should absolutely go to Europe and going in the winter is far more pleasant than the summer crowds.
I have been in Capri/Naples/Pompei in late January and it was cold! As it is the off season, some of the things we wanted to see were closed/limited hours and it was absolutely way too cold for anything beach related. If you wanted to visit central or northern Italy (Rome, Florence, Milan, Venice, etc) then it's fine to go in the winter. I lived there for a year and you get to see things without all the crowds or the summer heat. Venice often floods in December, so bear that in mind, but you will still have a wonderful time.
October is a fantastic month to visit Greece- probably the absolute best in my opinion, it's when I went. I did 4 days on a bus tour and 4 days independent travel to an island and then some independent time in Athens. Water in the ocean was still comfortable for swimming and the day temps were comfortable.
We have used this tour company for several day trips in Scotland and my friends used them for a full vacation in Ireland if you want to look into it for Europe:
https://www.rabbies.com/en/europe-tours/tour-italy
They keep a very steady pace and pack the day full, so bear that in mind...but you see a lot!
Another option is to go to a single central location and just visit there for a few days on your own pace (maybe doing day trips) before moving on to another location. Say you flew into Rome and visited there for a few days, then train up to Florence for a few days then flew to Athens for a week in Greece.
We have also taken trips in places where we simply hired a driver for the day to take us out to areas where we would not be comfortable navigating on our own.
There is no wrong way---just what is right for your traveling party at that time. I love Europe and think you should absolutely go!