Taking train from Modena to Cinque Terre

It could be done, but why? Modena to La Spezia is at least four hours each way, plus the time to get to and from the Modena station - - call it nine at best. The other train up the Liguria coast to Monterosso is about a half hour each way. That's another hour.
Ten hours down the tubes and you haven't done anything but ride trains. Why?

We are only in Italy for a week. We are there for a wedding and really only have two free days. We just thought it would be nice to see the coast. Would it be better to drive or can you recommend something else for our short time. We are staying in Modena

This can be done. And, quite easily. First, the suggested routing I'm about to suggest will be Modena/Parma/La Spezia Centrale. You will be on two different trains each way to La Spezia Centrale. At La Spezia Centrale, you change to the trains that stop at all of the CT towns. Second, all trains will be Regionale trains. For a Regionale train, you walk down to the station, buy your tickets, validate the ticket before boarding, get on the train and then find your own seat. No seat reservations permitted. It's like a City bus. Next, when you buy a Regionale train ticket in Italy, it has no train information on it. It's good for 60 days. You can ride any Regionale train making the run shown on the ticket up to 60 days. That's why you need to validate the ticket. Instant 50Euro fine on board if you are caught without a properly validated ticket. You will have one ticket for each train you will be on. Next, I suggest that you buy four tickets. Modena/Parma, Parma/La Spezia Centrale, La Spezia Centrale/Parma and Parma/Modena. You will also be buying tickets into and around the CT once you get to La Spezia Centrale. Next, I suggest taking the 6:35am train from Modena to Parma and the 7:48am train from Parma to La Spezia Centrale. You will arrive in La Spezia at 9:48am. Fare will be 15,85Euro one-way 2nd class. Validate each ticket just before your board. Don't validate all of them as once you validate, the ticket is only good for 6 hours (maybe 4 now). In the afternoon, do the reverse. Note, however, that the train to Parma runs about every 2 hours. I suggest either the 14:22 train or the 16:12 train. Parma to Modena is 16:30 or 18:30. Validate your ticket each time before boarding.

One more suggestion. Once you get to La Spezia Centrale, you can buy the tickets you need to travel from town to town in the CT. Fares are about 1.5Euro or so. Or, you can step into the TI office in the La Spezia station (if it's still there) and buy a 1-day Cinque Terre Treno card. Make sure it's the "Treno" card as there is a different card without the word Treno. This card provides unlimited train travel from La Spezia to Levanto for the one day. It also covers the tolls on the famous trails should they be open. Finally, you might want to know that you the same travel time distance to Venice via Bologna that you are to La Spezia Centrale. The actual distance is a bit longer but you would be on high-speed trains from Bologna to and from Venice. Reservations required on these trains but all of this comes with the purchase of the ticket. Fares would be double as it is more expensive to ride on the 160MPH train.

Thank you for the info. Do you have any other suggestions for us in our limited time

Sounds good how do I go about doing that?

Ginni, It is certainly possible to take a day trip to the Cinque Terre from Modena, but it will be a long day. The travel time will depend on which train you choose. In the same situation, I'd use a departure from Modena at 06:58, arriving La Spezia Centrale at 09:48 (time 2H:50M, one change at Parma, current fare €14.45 - DON'T forget to validate!). From La Spezia you can easily get to the five towns via the local trains (Riomaggiore is only about a 9 minute trip). For the return trip, there's a departure from Monterosso at 19:40, arriving Modena at 22:59 (time 3H:19M, two changes at La Spezia & Parma). DON'T miss that train as it's the last one of the day! I suggested that route, as that would allow you to start in Riomaggiore and then explore the other towns via train or hiking one of the trails, ending in Monterosso. The timing should allow you to have an early dinner before starting the journey back to Modena. I can suggest a restaurant with a patio overlooking the beach and also close to the station if you're interested (their Pesto Lasagna is excellent, IMO). While you could also have a look at the Adriatic side near Ravenna, it doesn't provide the same "ambience" as the C.T. and (IMO) wouldn't be as interesting. I was there in June, so it's still fresh in my memory. You can research your rail journeys on the trenitalia.com website. Note that tickets on Regionale trains MUST be validated prior to boarding the train on the day of travel. Failure to do so will likely result in hefty fines, which will be collected on the spot. Happy travels!

Thank you so much for all your info. I would very much like to have the name of the restaurant with the Pesto Lasagna

Ginni, The restaurant I was referring to was Cantina di Miky (unfortunately I think the website is only in Italian). It's on the main street only a few doors down from the train station. I usually dine there one or more times on each visit to Monterosso, and I was there again in June and had a great meal. You'll have to check the train times I listed above just prior to your departure, to make sure nothing has changed. Cheers!