Transportation from Modena to Lamborghini

I think your best bet is to just take a taxi with your time schedule. This recent forum discussion on the same topic might be helpful tho. Modena is a wonderful city, so the other two could enjoy exploring while you’re on your tour. That’s a long way to come, hope you enjoy it!
https://community.ricksteves.com/travel-forum/italy/transport-options-between-modena-and-lamborghini-museum

Hi, Giangilberto from Milan, I just checked on the Trenitalia website, and every morning there is a Frecciarossa train, they are high-speed trains, this type of train must be booked in advance because seats may run out and the price increases as the departure date approaches, which leaves from Bologna Centrale station and arrives at Modena station, I looked every day from now until October 15th, the train appears every day, so I really think it is always there:

Frecciarossa 8804

departure 08:18 from Bologna Centrale, arrival 08:38 Modena, journey time twenty minutes.
However, having said that, if I were you, I would contact Lamborghini directly, they are a company that has always had a care and courtesy towards the customer that makes the difference, and that is precisely what has made them great, not only the quality of the cars produced, but the service and the care with which they treat people.
To give you an example, I remember a very old interview with Ferruccio Lamborghini, the founder of the car company, in which he said that they offered a unique service even at the time, if your car broke down or otherwise had a problem that prevented you from continuing, you could contact them and they would send their mechanics wherever you were in the world, this makes you understand how much they care about their customers.
Now you are probably not the owner of one of their cars, or maybe you are, I don't know, but the difference is made by the service and if you have any problem and are unable to respect the arrival time for your tour I believe that they, upon contact, will be able to sort everything out.

Are you using the official site?

https://www.trenitalia.com/en.html

What is the date of this trip? If you provide the date we can be specific

The date is October 28th. I booked through Italo, but then it put us on a region train from Bologna Centrale to Modena with a 25 minute layover. When I look for the 8:18 train that was mentioned on that date, I cannot find it. Is it because the schedule hasn't been released yet? Or did they discontinue that route? I am not familiar enough with the train schedule to figure it out.

The 8:18 is a Trenitalia fast train and is on schedule for Oct 28
You won’t see it on Italo

Have you already purchased the Florence to Bologna leg?
If so just purchase the Bologna to Modena fast train on Trenitalia

If not I would purchase both legs on Trenitalia as a single journey so your connection is protected in case of delays

Then take a taxi from Modena to the museum

What is your travel date? I see arrivals as early as 8:21 am.

And if you are referring to the Lamborghini Museum in Sant'Agata Bolognese, why not rent the car at Bologna Centrale? It is almost the same distance as from Modena, and since the Freccia arrives at 7:28 am, that should give you plenty of time to get there by 9:30 am.

Please note that the Lamborghini factory and museum are in Sant'Agata Bolognese, a village located in the province of Bologna.
Public transport services are basically arranged by provinces in Italy, so is much easier go to Sant'Agata from Bologna than from Modena. Even by taxi is cheaper from Bologna, because when a taxi leaves the departure province the rate becomes higher.
There are even more car rentals in Bologna (being a bigger city than Modena), so makes sense travel by train to Bologna and rent there the car. A train that arrives at 7:30 give you enough time to do the papers for the rental and arrive in time at Lamborghini (better calculate about 1 hour from train station to arrive.
Lamborghini factory tour is done in a specific time: if you arrive later you miss the tour. For safety reason the tour has a time slot inside the factory, so people who are late lose it. Maybe they can join the next one, but only if there are spaces available (that it seldom happens).
@Giangilberto Do you remember that Ferruccio Lamborghini sold the Car company in 1972 because was losing too much money? In general the effort given to a customer ( = an owner of a Lamborghini car) is not the same of the one given to a museum visitor.

Thanks so much everyone. I ended up just buying new tickets from Florence to Bologna on Trenitalia. It was the only thing that really made sense. Will get the rental car in Bologna. Hopefully can get a refund from Italo for the other tickets.

Good lesson to always check both railways!

Beware of the parking situation at the Lambo factory. When I visited in July, parking was atrocious. There was no visitor lot on the premises (perhaps closed for construction?). We had to park across the street about two blocks down in a random packed parking lot. It was late morning, so I'm not sure if you'll have the same issue. Don't be picky. If you find something relatively close (street or lot), take it.

Hopefully can get a refund from Italo for the other tickets.

Well it is only 9 EUR for 2 people each way.

@ricky, I have an in-depth knowledge of the company's history, I work in photography and have actually done work for them myself. While the company was indeed sold to two Swiss entrepreneurs, it wasn't because of issues related to customer care. There were three main reasons that forced Ferruccio Lamborghini to sell: the first was a major crisis in the tractor sector, the second was the socio-cultural climate that defined Italy from the late 1960s through most of the 1970s, a period known as the "Years of Lead," characterized by great political turmoil and frequent strikes in many factories.
The working class had united and was demonstrating, consequently, the luxury market was hit hard, not least by the energy crisis. All these factors led to the sale of the company in order to save it.

As nixit71 says, the parking could be a problem, mostly during the week in normal working time.
Due to the expansion of the Lamborghini factory and being in the middle of an industrial district when all workers are in the streets are almost sold out.
Inside the factory area can park only customers and managers.
The visitors parking lot should be the one between via Modena and via Costa, but is small and immediately filled by the workers.
Via Turati, via Costa and via Lamborghini are the streets where usually can be find a slot where to leave the car (beware where are time limits!). I never tried near the graveyard, that isn't far. The big parking lot on via Modena (opposite side of Lamborghini) is for workers only, so available only during the week end when the factory is closed.

I don't know if the times work, but we took a bus from Bologna to the factory. It dropped us off about a 5-10 minute walk from the factory entrance. I want to say the bus ride was 1 hour. If you don't need the car for any other reasons that might be an option for another way to get there. The factory tour time is not flexible, I'd try to get there by at least 9:15 (probably earlier if you need to deal with parking) as they started exactly on time when we were there.