What is the best way to cut a 20 inch vine (font) monogram with a circle around it, out of 1/4 inch Baltic birch plywood? I have seen them everywhere but I'm finding it extremely difficult to cut out with my jig saw. Are there better tools and/or techniques available that would make it easier?
A long neck scroll saw is the animal of choice. It uses short strokes and you match the blade type to the material.
Heres a short discussion of the types of scroll saw blades
If you want to save money and space (but not time) over a powered scroll saw, another alternative is to get a jewelers saw or a fret saw. These are essentially coping saws that are smaller and in different shapes. Some have very deep throats that help to access the middle of a large piece. Like with a powered scroll saw, you can take the blade on and off to access and cut fully enclosed holes. This method requires good clamping too, preferably with the board vertical in front of you at about workbench height.
Oh, and be sure to read reviews before you buy. There are many hand-powered tools out there that are worse than worthless, but also plenty of good ones.