We are doing that as well this summer and will probably take a taxi to the train station, although we might walk depending on time - seems to be about 1-1.5 miles and a 20 min walk based on my research, although I haven't actually done it yet.
Our plan is to take the train to Riomaggiore and then take the ferry boat up to Vernazza, then on to Monterosso and spend some time there having lunch and at the beach.
Looks like the train only leaves at 8:12, 9:51, 10:06, 11:10 - although that is for a day next week, I don't know if it switches up in the summer, but has been those times only for the months I have been checking. Our plan is to get there for the 9:51 or 10:06. We didn't want to take the ferry all the way there, since it's like almost 2 hours - too long and I don't think I will make it on a ferry for that long.
Links:
This is the site for the CT ferry service :
http://www.navigazionegolfodeipoeti.it/orario.asp?lang=eng
And an old helpful thread:
https://community.ricksteves.com/travel-forum/transportation/la-spezia-to-cinque-terre