You'll need some sort of warm layer and a waterproof outer layer, for sure. All of those places are likely to feel sort of damp at that time of year, which adds to the chill factor.
Underneath, I'd probably wear long sleeves, and I'd travel with a set of lightweight long johns. But I'm quite cold-natured. I'd also want something warm I could put on my head (though I might not need to use it) and gloves. Depending on the neckline of your warm layer and waterproof layer, a scarf might be a good idea.
One thing to keep in mind is that you may encounter some very warm interiors, so give some thought to how you'll cope when you have to start peeling off layers.
You can take a look at actual day-by-day historical weather data (much better than monthly averages) at wunderground.com. After entering your target city, choose "History" and enter April 1 of a recent year. Scroll down just a bit and click on the "Monthly" tab. Scroll down some more for a graph of temperature over the course of the month. Repeat for other years to get a better idea of the range of temperatures you might expect.
As you get close to your departure date you can use your favorite weather website (or wunderground) to see what's happening at your first stop.
Edited to add: I've just remembered that people with off-season experience in that area have discouraged dependence on umbrellas for rainy days because of the wind. They suggested a hooded waterproof outer layer instead.