How do you say
"I think I understand what you mean" in Japanese. I have started speaking to a native Japanese speaker, and I want to be able to assure them that I'm getting what they mean. Also, how would you express that you feel just as nervous and unsure to someone. Like how would you say "I feel embarrassed and unsure about my Japanese, so don't worry about your English. We are both learning, and this is just part of the process." Something along those lines. A simple way to express that thought.

Question

Answer

"I think I understand what you mean"
--> あなたが言いたいことはわかります(よ)。/ あなたがなにを言いたいのかわかります(よ)。

"I feel embarrassed and unsure about my Japanese, so don't worry about your English. 
--> 私も日本語に自信がないし、日本語を話すのはちょっとはずかしいです。だからあなたも(あなたの)英語のことを心配しないでください。
We are both learning, and this is just part of the process."
--> ふたりとも勉強しているのですから。(恥ずかしいのは) 最初だけです(よ)。

note: These translations are only "my opinion", I think there are various ways to say them. I hope this helps you.