How can I get my redstone to activate adjacent pistons?

There are actually a large number of different ways you can do this. Perhaps the simplest is using redstone repeaters, like so:

There's also the design proposed by StrixVaria, which uses elevated redstone:

But there are even more ways you can do this. Here are a few:

Take your pick!

Try this:

a---+++
    ppp

• a - pressure plate
• - - redstone
• + - redstone on a raised block (the block should be at the same level as the piston)
• p - piston

In order to power pistons, you either need the redstone line running directly into the piston, run the redstone line on a raised block, or using redstone repeaters. You can also power an adjacent block using a repeater, which can be very useful for certain applications.