I sometimes get doubts about Islam out of nowhere and some of them are very nonsensical things. Should I ignore these doubts or try and resolve them through research and logic?
Doubts come from our worst enemy, Shaitan. Shaitan wants to disturb the believers and deceive us. Doubts should never harm the believer who thinks in a proper way. You write that some of the doubts are very nonsensical. This means that you know that it is nonsense. Such nonsensical doubts need no effort to refute as you know that it is nonsense. Any doubt which you think it disturbing you, needs to be refuted with evidence. You can ask authentic scholars or refer to authentic books or websites.
We in Shia Islam don't have any question with out proper answer, and no doubt is left with out logical reply.
Wassalam .