Milan Linate Arrival Logistics

There are no tracks at Linate and high speed trains to southern Italy leave from Milano Centrale, you don't have many alternatives. It's a 20 minutes bus ride, nothing to be worried about.

I was wondering if we had to get to Malpensa first, but it sounds like we can take a bus straight to Centrale?

You can take a bus to Milano Centrale.

Linate is only 5 miles from Milano Centrale (instead of Malpensa, which is 30 miles away).
You can take the non stop Airbus for 5€ or a taxicab for about 5 times as much (no flat rate, goes by the meter). Transfer time is under 20 min by taxi, or about 25-30 by bus. It's up to you, if there are a few of you a taxi is a good deal.

Good to know! We are a party of 4 so a taxi might make sense.

Yes, you have to go to Centrale to take a train to Florence.

A possible option could be a taxi to Milano Rogoredo station, that is quite close to Linate and has some services to Florence, both on Trenitalia and Italo. But services are not as frequent as from Centrale - if you hit the right timetable you may spare one hour going through Rogoredo, if you hit the wrong hour it could take more time. I leave this hint to professional travelers but first timers should better head to Centrale :-)

Thanks all! The taxi was €17 from Linate to Milano Centrale, 15-20 minute ride.