disclaimer: I’m not ethnically or religiously Jewish (I am a Christian that practices my faith in a very “Jewish” way)
Honestly, it depends on who you ask. Some Jews will not turn any lights on, start their car, push an elevator button (if you go to Israel there is a Shabbat setting on the elevator so that you do not have to press the button), etc…
For me personally, I do my best to focus on spending time with the Lord & with family. We start on Friday night by rejoicing in the Shabbat meal together. Saturday throughout the day we have service or will watch teachings & continue to spend time with family. Spending time with & giving praise, thankfulness & worship to the Lord is the focus. We still cook on Sabbath, we go out of the house, exercise, etc., however we do not go places & partake in activities that require other people to work (restaurants, etc.) so that we are not requiring others to work for us.
In regards to the verses referenced in the NT about picking up mats, that was a violation of rabbinical Judaisim’s man-made laws. To keep it simple, the Bible says to rest & not work on the seventh day (Exodus 20:8-11), so the Pharisees created “extra” laws to ensure a person would not even come close to “working.” (This was also a way to “control” the people & remain in “power”) Think of the Southern Baptist view on alcohol. There is nothing against drinking in the Bible, in fact it is encouraged & instructed in various places, but I think we can all agree that it’s not meant to be taken to a level of “drunkenness” or “too far.” However, we all know the “zero tolerance” policy of some (not all) Southern Baptist groups. The idea is that if you never drink it, you can never take it too far. And unfortunately, the commandments of the Lord can be disregarded & while the traditions of men are held as doctrine (Mark 7:7-9). Hope this example helps. If you were to look at books such as the Talmud, you would find that there are nearly 500 “extra” laws pertaining to Sabbath that are not found in the Bible. A quick example is Shabbat 108b where spit is not permitted in a person’s eye on a sabbath day. Well, guess what day it was when Jesus rubbed spit (mud) in a man’s eyes to cure his blindness… it was sabbath. It was also not permitted to heal on sabbath or to spit in the dirt (this creates mud which was considered “working”). So Jesus was “breaking” the man-made laws & traditions because they violated the Lord’s command not to “add or take away from” the commandments given to Moses (Deuteronomy 4:2)
I hope this wasn’t too lengthy & answers your original question.
