There really is no cheap way to get from Milan to Lauterbrunnen. It's the higher-speed trains that run between the two countries on this north/south run. It will cost about 52Euro from Milan to Spiez where you will get off the high-speed for the run into Interlaken. From Interlaken Ost, track 2A, you will take the local train into Lauterbrunnen. The local train from Spiez to Lauterbrunnen will cost 17.20CHF ($15.00).
A few of hints: 1) for what you are describing, you don't need a Eurail pass for either Switzerland or Italy. Did you know that in Switzerland, your Eurail pass only covers 25% of the fares from Interlaken into Lauterbrunnen Valley? This 25% limit includes everything in the valley including Wengen, Gimmelwald, Murren and travel to the two mountain tops. P2P is less expensive. If you plan on going to the two mountain tops, a Swiss pass or card of some type is better. 2) Did you know that in Italy, they charge Eurail pass holders an extra 15E-20E reservation fee for riding the Eurostar-class trains? You will be riding these trains from Rome/Florence, Florence/Venice, Venice/Milan. P2P is less expensive. If the P2P fare between Rome and Florence is 39.10Euro, why would you pay for your pass plus 20Euro to ride something that costs 39.10Euro. You don't need a Eurail pass in Italy. 3) You will most likely purchase your ticket from Milan to Lauterbrunnen in Italy. The Italian train system cannot see Lauterbrunnen on their screens. They can only get you as far as Interlaken Ost (East) which is the train station you want. The Italians cannot see the Jungfraubahn or Schilthornbahn railways on their screens. It's OK. Buy the ticket to Interlaken Ost. Buy the ticket from Interlaken Ost to Lauterbrunnen when you get to Interlaken Ost.