Have you looked into drop off charges for renting in Austria and dropping off in Italy? They may be so high as to make the train a good bit cheaper. Also, gas is US$4.50 a gallon here, and as noted above, parking is hard to find and expensive. Also, as noted above, you do not need and in fact cant really use a car in Grindlewald and cant use it at all in Zermatt, so you would be paying for parking and for the car to sit idle for a few days.
There are numerous excellent walking and hiking opportunities from the Grindelwald area and Zermatt, so you would not need to drive to get to good hiking, another reason not to have a car.
I agree that for the most part, the views from the train are just as good and in some cases better, than views you would get from a car. The trains can often be at higher elevations and more isolated areas than the roads, esp the dual carriageways.
I dont like driving in Lucerne as it is quite congested, and prefer to take the train in (and I live in Zurich and am used to the confusing maze of one-way and pedestrian-only streets and the lack of good signage, not to mention driving on the same roads as the trams, which is a little unnerving until you get used to it . . .)
If you have a car, avoid driving to Lugano or into Switzerland from Austria on a Friday or Sunday, as you will run into weekend traffic, this is esp true for the Gotthard Tunnel into the Lugano area.
All the above notwithstanding, if the drop off charge is reasonable and you dont mind paying for gas, parking, and for days you are not using the car, there are some good drives. Most esp are the passes like the Brunig, Susten, Grimsel, Furka and St Gotthard passes. The Grimsel pass is a good way to get over to the Zermatt area. All but the Brunig pass are significantly above the tree line and offer interesting views; at the Furka you can stop and get fairly close to the glacier which is one of the sources of the Rhone. I would say that the Furka is not for the feint of heart, but the other passes are not hard to do (have not yet done the St Gothard myself, but that is what I understand from others).
Depending on where you are coming from in Austria, it may not be too out of the way to drive through the Engadine Valley to St Moritz and then over the Julierpass, which has some great scenery; however IMO the prettiest part of the trip is by train between Pontresina and Fillisur on to Chur, but the route by car is totally different so you dont get the same scenery.
Of course, you can do all these passes by the public Swiss Post Bus, they just take more time and you have to work out your schedule to their timetable, and you may need to switch a few times to get to Zermatt from Grindlewald via Post Bus (or take trains for part of it). You could also rent a car for a day or to just in Switzerland and just to do some of the passes, as I mentioned above, the drive to Zermatt from Grindelwald via the Grimsel pass would be a great drive, albeit a longish day in the car. (there is also a car train through the Lotschburg Tunnel, but what is the point of that if you want to see scenery?)
