First, correction.
我要多学习 (it's ok, but it sounds not very native. It seems something is missing here.) => 我要多多学习。
OR 这书写得很好,我要多学习(这本书的内容). 我要多学习 needs an object!. 我要多学习。你要多学习啥?
我要更学习 (it is NOT correct.) => 我更(加)要努力得学 OR 我要学得更(加)努力。
更 is used to strengthen Adj. or Adv., not Verb.
For example, For Adj. 她那件裙子我觉得更漂亮 (I think her skirt is more beautiful.) For Adv. 我跑得比你(更)快。 I run faster than you.
--------------------------------------------------------------------------------------------------------
In a word, 多 is like many (generally) or more (in quantity, many 的比较级) or more often (just like the first case you mentioned). 更 is like more (+adj./adv.) in English.
--------------------------------------------------------------------------------------------------------
Back to your question, how to translate "I need to learn more."
我得多学点。( I think you probably want to express this meaning. It implies or emphasizes that I need to learn more things not in time (more often) or not more hard but in quantity.)
Besides, 我要 = I want (In German, Ich will. If you can German, it would help. If not, it is ok.)
我得 = I need to (becasue something for example pressure drives me, and I should or have to do something)
For example, 今晚我得早点睡。I need to go to bed early tonight (maybe I need to go to work or school tomorrow)