>> How long does the train stop at each station?
If you are boarding at Zürich Hauptbahnhof (main station), the timetable indicates that trains heading to Luzern will stop there for at least 5 minutes, sometimes 10.
On the platform, you will see blue-and-white overhead signs with the letters A, B, C, D etc. These denote platform sectors. The overhead electronic display on the platform showing the destination and departure time for the train will show at which sectors of the platform the 1st and 2nd class carriages will be when the train pulls in. You can save time by waiting on the platform at the appropriate sector before the train arrives.
Luzern will be the end of the line, so you will be able to alight comfortably. In any case, as far as I can recall, the conductor stands on the platform at every station to ensure that all passengers have completed alighting and boarding before giving the signal for the driver to pull out.
>> What's the best time to travel between Zurich and Lucerne?
Avoid the times that people will be travelling to and from work, and make sure to get a direct connection, so you don't have to change trains en route.
>> Will we be able to carry 4 bags into and off the train or should we send our luggage to Lucerne by travel services? As my wife will be carrying the baby I would have to manage the luggage alone..
How many of these 4 bags are suitcases, and what size are they? Are you carrying any backpacks?
Lucas has given you a link to explain what luggage services are available. If you don't want to pay the additional fee for Express Luggage, you could consider packing one suitcase, and one backpack (if you will have one) with what you will need for the first couple of days, and taking that with you on the train. You could then send the other bags via the standard SBB luggage service and pick them up 2 days later.
If part of your luggage problem is that you have to take business attire etc (as you mentioned this is partly a business trip), you could consider packing all that stuff into one bag, leaving it in the Left Luggage at Zürich, and picking it up when you return to Zürich at the end of your holiday.
If I remember rightly, the trains from Zürich to Luzern are double deckers, with level entry from platform to train. If so, that will make things easier for you.
It seems that all the trains on that route also have a Family Zone. These trains are denoted in the timetable as "FA". The Family Zone will be a 2nd class carriage, and will be either the first or last carriage on the train. If first class carriages are at the front of the train, the family carriage will be the last carriage and vice versa. These carriages have extra room for strollers.
This article might also be useful: