I'm thinking of taking a train down from San Francisco to Los Angeles in early March. It's probably going to be one of those overnight trains, but I'm not entirely sure of my plans for method of transportation yet. Do you think I can just show up at the train station and purchase a ticket or would I have to book in advance? How likely are seats going to sell out?
If you are referring to the Amtrak Coast Starlight, it is an all-reservation train. While it may not sell out in March, capacity is limited so I do recommend booking in advance. Do see the website Amtrak and click on the DEALS tab on the top of the page.
This train will most likely be packed in the summer months, but if you do wait to book, you should have no problem getting on it in March.
You should be aware, however, that this route only has one train per day--so booking early is recommended if you do a time constrained trip where you have to travel on a certain day.
The scenic part of the Coast Starlight route is along the Pacific Coast between Santa Barbara and San Luis Obispo, and best viewed on the northbound run. I believe it will be dark when you pass that area on the southbound train. This train is frequently quite late - as much as 11 hours late for me last year (the freight railroads own the track and only allow Amtrak when they're not using them).
For the past few years, many more people are using this service to travel since it's considerably cheaper than current airfares. Yes, you should reserve for this train. Amtrak does not service SF at all - they run a bus from the city to the East Bay station at Oakland. The only trains running out of SF are the BART lightrail and CALTRAIN to the Silicon Valley and San Jose.