You can book from Milano Centrale all the way to Lauterbrunnen on SBB. That will save you the trouble of buying separate tickets or buying them at the station in Spiez.
If you want to make the trip as simple as possible, I suggest you book the routes with the fewest changes. Otherwise, you could have 3 train changes between Vernazza and Milano Centrale, and another 3 between there and Lauterbrunnen. Also, if you book the journey that goes from Milano to Spiez without a change at Brig, you will get a better price on SBB ( assuming you book now for September).
Example: comparing 2 routes to Spiez from Milan, one with a change at Brig and one without, I see for a random date in late September the following:
With a change at Brig: normal price 91 CHF, Special Offer price 81 CHF..
For the direct EC train Milano to Spiez (EC 54): normal price 91 CHF, Special Offer 49 CHF ( a day later is is 55 CHF).
The simplest journey, with the fewest changes, is to depart Vernazza at 10:34 (a nice civilized hour) and change at Levanto to the IC 666 which goes all the way to Milan with no change at Genoa. ERlier departures from Vernazza will have more changes, and May route you through Firenze instead of along the coast. Your choice. Either way, you buy this ticket on Trenitalia.
If you take the IC666 option, you arrive at Milano Centrale at 13:53. The direct train to Spiez, which I mentioned above, departs at 15:20. This is a lot of time to spend in the station, but you can get lunch, etc. I do not see other, earlier, options offered on SBB, but there may be something if you go the other way, via Luzern. I’m did not explore that option. The direct train to Spiez is the simplest and appears to be the least expensive, but it does involve some tunnels if that matters.
However you get to Spiez, you will use Swiss regional trains to reach Lauterbrunnen, with a single change at Interlaken Ost. That one is unavoidable.
If you are tunnel-averse like I am, the route to Lauterbrunnen via Luzern can be scenic and nearly tunnel free, but the route over the Gotthard Pass instead of through the Gotthard Basistunnel tunnel will take longer and takes careful planning on SBB.