I don't think you necessarily have to try TAing, but I would say it would be very valuable to get some experience in a UK primary school setting to see what it's like with the mix of subjects and larger classes. It would also probably be valuable to get some experience in a UK secondary - as if you have just been teaching small groups, then you will likely find the actual classroom experience quite different.
In order to get experience, you can just email schools and ask if you can observe for a few days. You can also contact Get Into Teaching, as I understand that they can help organise school experience.
My personal view is that if you're mainly interested in teaching because you have a passion for your subject, secondary is probably the way to go. I teach secondary maths and I do think there is a fair bit of variety in the day - my year 7 students are still quite playful as they've just come from primary school, but my older students can have good intellectual discussions.