I've come across a wild Pokemon with a red circle around it and it broke out of the Pokeball rather quickly. Is there something I can do next time to increase my chances of capturing it?
Take a look at this article Pokemon GO: 9 Essential Tips, Tricks & Answers To Your Questions
How to capture Pokemon
Use the Item bag to toss Pokeballs or use items (like Razz berries) to make capturing a Pokemon easier. Improved Pokeballs such as Great Balls, Ultra Balls or Master Balls will also improve your chances.
The targeting circle tells you how tough the capture will be. A green circle is best, following by a yellow, orange, and red circle. Yellow circles, in our experience, are a pretty tough capture — orange and red? Wait until you’ve got the right items to make capturing easier.
Wait until the targeting circle is at its smallest before flinging a Pokeball. The smaller the circle, the better your chances of capturing will be.
Yes, as you progress further into the game you will find Berries which will make it easier for you to catch Pokémon during encounters.