new to Switzerland and northern Italy

1) Taxis in Basel are expensive, and I am on a Swiss salary and I say that.
There are 3 possible places boats dock in Basel: St Johann, Klybeck and Dreiländereck. A taxi from any of them to "Basel SBB" station will cost CHF 40-50. There are always taxis waiting when the boats dock.
Taxis are normal size cars, you will need two. I haven't seen any taxis big enough for 6 people in Basel.

Basel is a tram-city, everybody travels by tram. Tram costs CHF 3.80 per person.
If you are landing at St. Johann the tram is easy, tram stop is a short walk across a park. The other 2 are in the middle of the docks, 5 minute walk to the nearest tram stop. Possibly this justifies a taxi if you are a group.

And the luggage issue. You need to carry your bags onto the train and stow them yourselves. All luggage is "Hand luggage" Do not bring more than you can schlep yourself without assistance.

2) If you are staying on the east side of Lake Como (Varenna etc.) the Bernina route makes sense. Unless you are starting early from Basel you need to overnight en route.
Other parts of the lake are not accessible from the Bernina route.

Val et al, we did the Viking cruise last fall. Note that one of the things they do, the day before the cruise ends, is ask for people who are traveling onward (and not on their package extensions) to let them know if they need taxis, to where, and how many people. Then they will have a taxi ready and waiting for you, by name, and the appropriate size, at the dock upon disembarkation. No need for you to fret about or book in advance - its their standard procedure. Super easy. In our case, the six of us went to the SBB train station where we bought tickets and got on a train. There are two train stations so you need to specify.

What town on Lake Como? It might make a difference on how easy it would be to get to. I think backtracking is seldom a good idea.

In Milan, the number one attraction is a viewing of DaVinci's Last Supper. You need tickets in advance and getting them is a whole new subject area. Milan is a big city with the usual assortment of city attractions - museums, shops, the Duomo, etc.

Dont forget they use Swiss francs in Switzerland not euro.

A few thoughts on your questions......

1. No, stick with well planned rail trips as that will be far easier. Keep in mind that if you rent a car in Switzerland and drop in Italy, there's usually a steep charge for a one way rental. Each driver listed on the rental form will require the compulsory International Driver's Permit for driving in Italy, and you'll have to deal with the ZTL (limited traffic) zones in Italian towns. Each pass through one will result in hefty fines. There's also the issue of parking, tolls and speed cameras.
2. In order to know whether the Bernina Express would be a good option, it would help to know which town on Lago di Como you'll be staying in. Varenna is the usual favourite with many here and that's an easy solution if taking the B.E. Taking a day trip on the B.E. may not be feasible because of the travel times involved. It would be a LONG day.
3. Again, without knowing where you're staying, it's difficult to offer suggestions. You might have a look at the RS Italy guidebook.
4. The guidebook has lots of suggestions for things to do in Milan. You could have a look at the Duomo, Sforza Castle, the Galleria Vittorio Emmanuele, do some shopping at La Rinascente stores or see The Last Supper (reservations required). Three days in Milan would be a bit too long for me, but if it's your first time there you should be able to find enough to do for that time frame.

One additional comment related to #1 above...... there are also some potentially expensive caveats to be aware of when using trains and other public transit (ie: validating locally purchased tickets on Regionale trains or Metro). If you're not aware of those, be sure to do some research on the subject.