There's the Oregon Rail Historic Center, of course (http://www.orhf.org/), just across the street from the Oregon Museum of Science and Industry. You can get there by the Portland Streetcar from downtown.
Then bring a picnic lunch to The Fields Neighborhood Park in Portland's Pearl District at NW 10th Avenue & Overton St., just three blocks north of Union Station, and a block north of the NW 10th & Northrup Streetcar stop. The benches and picnic tables in the three-acre park give you a front-row seat to nonstop freight and passenger trains on the main between Portland and Seattle. Each day there are four Amtrak Cascades trains each way, one Coast Starlight each way and the Portland section of the Empire Builder each way. Though this is BNSF (formerly Spokane, Portland & Seattle) trackage there are frequent UP freights shuttling between UP Albina Yard and BNSF Lake Yard, plus one Portland & Western freight each way every day. The park itself is built on what used to be the SP&S Hoyt Street Yard.
There's a great Moroccan coffee shop right next door, too.