Traveling from Zermatt to Como

Unless you carefully plan your route through scenic detours, there will be a lot of tunnels in both routes, more so going via Zürich.

If you want a scenic trip, you should plan a route Zermat-Visp-Andermatt-Göschenen-Chiasso-Como, which comprises half of the "Glacier Express" route and half of the "Gotthard Express" route. Alternatively, you can go from Zermatt to Domodossola, and then take a train on the Centovalli railway to Locarno, and from there use regional trains to reach Como.

In both my suggestions, travel time would be substantially longer but you'd go through some scenic routes.

The route via Goeschenen takes 7h20min (I recommend that you put a break of couple hours in Andermatt for lunch). The route via the Centovalli railway takes just short of 6h and the fast routes via Milano take 5h20min.

Traveling via Zürich takes 6h20.

The travel agent is talking rubbish. The train from Zermatt to Zürich is not high speed, it is normal speed.

not on premium trains that we could make reservations for

Translation: not on trains I can make a profit on by selling you reservations. You do not need reservations on Swiss trains. You do need reservations on Italian long distance trains (not on local trains). But when you buy a ticket, it comes with reservations if needed. You only pay extra for reservations if you have a rail pass, which is not good value in Italy anyway,

First question, where on Lake Como do you want to go? Is it Como town, or somewhere else on the lake? There is a rail line up the east side of the lake going through places like Varenna. If you want to go to Bellagio, or somewhere on the west coast like Menaggio, you will need to take a bus, and a totally different route.

You can look up train times on the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en/
This will always find you the quickest route, and any route from Zermatt to Lake Como involves crossing the Alps.
For example, I did a quick check for a random date in August, Zermatt to Varenna-Esino. Lots of options, including but not limited to:

Zermatt depart 08:13, Brig arrive 09:37
Brig depart 09:44, Milano Centrale arrive 11:37
Milano Centrale depart 12:20, Varenna-Esino arrive 13:23

Note, this is going nowhere near Zürich, that would be a huge time consuming detour. It is going via Stresa, but you don't get off there.

Finally, how are you getting to Zermatt? And from where? You need to make a considered decision whether you buy a Swiss rail pass, and if so which sort. If you have a pass, you are covered to the Swiss border.

We are staying in Bellagio. She told us we would need a water taxi from the train station to get there after our rail journey. We have already purchased the Swiss passes because before going to Como, we are also visiting Lucerne and Interlaken!

dbrother, If you are staying in Bellagio, you can do the route I described (Zermatt - Brig - Milano - Varenna) and then a ferry boat (not taxi).
I suspect there are other options, like:
Bus from Como town (Zermatt - Brig - Milano - Como town - bus)
boat from Menaggio (other side of lake: Zermatt - Brig - Domodossola - Lugano - Bus to Menaggio - ferry boat)
Both those would need research.

If you haven't seen a map of the lake yet, these might help: https://upload.wikimedia.org/wikipedia/commons/6/60/Lago_di_Como.png
If you have a Swiss pass, that is valid to Domodossola, which is the last Swiss railways station on the Brig - Milano line. You then need to buy a ticket from Domodossola to Milano, and Milano to Varenna. The first will include a reservation, the second doesn't.
See map here: https://www.sbb.ch/content/dam/sbb/de/pdf/freizeit-ferien/ferien-kurztrips-schweiz/internationale-gaeste/sts-geltungsbereich_en.pdf
You train from Brig to Milano will stop at Domodossola, but you don't need to get off, so long as you already have a ticket from Domodossola to Milano.

This is all great information! I will make sure to go one of these routes instead! Thank you so much! Cheers :)

I see you have Interlaken in your plans. Most of us would recommend you go beyond Interlaken (a rather ordinary town) and stay in the Lauterbrunnen Valley, either Lauterbrunnen itself or the scenic mountain towns of Wengen or Mürren, where you can see the mountains and get to them more easily.

The train ride from Milan to Como goes through some poverty ridden neighborhoods most of the time. Graffiti all over. The train itself had a hole in a seat to the tracks below for the WC. This was in 2015. There is no scenic beauty on this train ride until you enter Como.

Bellagio is where the two arms of Lake Como split, right in the middle. Lake Como is shaped like a wonky upside down "Y" and the two arms are quite long. Bellagio is at the fork, in the centre. It has no train lines. Varenna is nearly opposite Bellagio on the east shore, has a real train station (unmanned), is absolutely gorgeous, and the ferry dock is quite close to the station. There are frequent ferries which take just a few minutes between Bellagio and Varenna, both passenger ferries and passenger and car ferries. That's the route I would suggest, and it would involve changing from the Swiss train at either Milano Centrale or at the suburban station of Monza. Opposite Varenna is Menaggio which is linked to both Varenna and Bellagio by ferry, and all of them have ferries which go down the western arm to Como and lots of other stops. It is quite a long way by boat from Como to the mid lake region where Varenna, Bellagio and Menaggio are, but ferry from Como is possible, both the normal ferry and the hydrofoil (which has relatively bad visibility and you must stay inside).

I have just taken this route from Zermatt to Lake Como. We actually spent two days in Stresa and then travelled on to Varenna via Milan. We had a Swiss travel pass, so had to purchase a ticket at Domodossola (Swiss/Italian Border) for the journey to Stresa. we stayed in Stresa for two days so the afternoon before our travel to Varenna (then onto Tirano for the Bernina Express) we went to the train station in Stresa and purchased our follow on Italian tickets. The women in the ticket office was extremely helpful. I was a little nervous changing trains in Milan but it was simple. We literally walked two platforms over for our next train. Once in Italy the scenery is not great from the train as someone else mentioned it does go through some impoverished areas with graffiti . However, once you get to Lake Como and Varenna it is so beautiful. (the train station is Varenna Esino and direction Tirano). You can easily get the ferry across to Bellagio. (15-20 mins). Always remember to validate your Italian tickets - its a little box inside the station or right on the platform. Simple slide the ticket into the machine until it punches a date/mark.