The basic meaning of the verb vivere is “to live.” In this case it’s intransitive, meaning it doesn’t take a direct object.
Non è per niente male vivere in Italia, anzi!
It's not at all bad living in Italy, on the contrary!
Caption 54, Francesca - sulla spiaggia - Part 3Play Caption
But Italian also uses vivere to mean “to go through,” “to experience.” In this case, it’s transitive, meaning it takes a direct object. In this first example, Gualtiero “lives” the problem (direct object) of having to eat out every day.
Quindi ho vissuto in prima persona il problema del pranzo fuori casa.
So I experienced firsthand, the problem of lunch away from home.Play Caption
In talking about how he experienced this problem, he could have said, ho vissuto personalmente, but he used in prima persona (in the first person, firsthand) which is a very common way to say the same thing.
As a guest in a foreign country like Italy, you'll often be asked about your esperienze (experiences). If you use the noun form, then esperienza is your friend.
È stata una delle esperienze più intense della mia vita.
It was one of the most intense experiences of my life.Play Caption
When instead you want to use the verb “to experience,” then vivere is a good choice:
Nel ristorante che stavo ideando, la cucina e l'ambiente stesso avrebbero dovuto risvegliare emozioni da vivere e condividere con il cliente.
In the restaurant I was designing, the kitchen and the space itself would have to awaken emotions to experience and share with the customer.
Captions 14-15, L'arte della cucina - L'Epoca delle Piccole RivoluzioniPlay Caption
And here’s Jovanotti in his song about positive thinking, with a verb and a noun that both mean “experience”!
...e vivere le esperienze sulla mia pelle...
...and live out the experiences on my own skin [personally]...
Caption 32, Lorenzo Jovanotti - Penso PositivoPlay Caption
It wouldn’t sound good to say “experience the experiences” in English (using “experience” as both a verb and a noun in the same line), but it wouldn’t be incorrect! And now, you've discovered still another way to say “personally.”
In a nutshell:
There are two important (related) meanings of vivere:
vivere (to live)
vivere (to experience, to live out, to go through)
There are different ways to express “personally.” Both Gualtiero and Jovanotti could have used any of the following to mean pretty much the same thing:
sulla propria pelle (on one’s own skin, firsthand, personally)
in prima persona (in the first person, firsthand, personally)
“Experience” as a noun is pretty much the same as esperienza. There’s no verb form of esperienza.
“To experience” is most frequently translated as vivere. Check out some of the other possibilities here.
Just for fun:
Non sono italiano ma vivo in Italia. Vivere qui è un’esperienza straordinaria. Hovissuto dei momenti fantastici, però ho anche vissuto sulla mia pelle cosa vuol dire essere extracomunitario. La procedura delle impronte digitali l’ho vissuto in prima persona. Non è stata un’esperienza per niente simpatica, e personalmente, ne avrei fatto a meno di viverla.
I’m not Italian, but I live in Italy. Living here has been an extraordinary experience. I’ve experienced some fantastic moments, but I’ve also experienced, firsthand, what it means to be non-European. The procedure for fingerprinting I experienced firsthand. It wasn’t a nice experience at all, and personally, I could have done without going through that.