Learn Russian: Я согласен.

No, in Russian, согласен is actually a 'short adjective' meaning 'in agreement'. The sentence literally translates to 'I [am] in agreement'. The word for 'am' is simply left out in the present tense.

Do I need to change the ending of согласен depending on who is speaking?

Yes! Because it acts like an adjective, it changes to match the speaker's gender. A man says Я согласен, a woman says Я согласна, and for the plural 'we agree' you would say Мы согласны.

How does this relate to the simple future tense (perfective verbs) we are learning?

This specific sentence is in the present tense. However, to say 'I will agree', you can use the perfective verb согласиться. When you conjugate a perfective verb with present-tense endings, it creates the simple future tense. So, 'I will agree' becomes Я соглашусь.

Can I just say 'Я буду согласен' to mean 'I will agree'?

You can, and it literally means 'I will be in agreement'. However, using the perfective future verb Я соглашусь is much more natural and common when you want to express making a decision or promising to agree to something.