I need to analyse data and am unsure whether or not I should use a multinomial logistic regression analysis or a binary logistic regression analysis. I have just conducted a latent class analysis and am interested in seeing if my variables can predict class membership. There are only two classes and so my response variable only has two categories; however the variables range from having 2 to 3 to 4 and are mostly nominal categorical data.
I think that, if a model's outcome variable has only two categories, then the appropriate term for the analysis is binary logistic regression, regardless of the number and type of predictors. For details on performing actual logistic regression analysis, please see links in my recent related CV answer as well as this tutorial, this chapter (SPSS examples) and this nice set of slides (Stata examples).
In regard to potential multicollinearity, if you're not familiar with this topic, I suggest you start by reading corresponding Wikipedia article. Also, check some CV discussions, for example this, this and this. If you have access to paywalled peer-review journals through your institution, you might want to check this paper and this paper. This blog post presents some options on dealing with collinearity. More details and references on the topic of multicollinearity in general (not specific to logistic regression models) can be found in my recent relevant answer. I hope that this is helpful.
Do realise that it is of importance to know when you have more than two outcomes, if the outcomes are ordered or unordered. In your case (class membership y/n) I don't understand how you can have 3 outcomes unless outcomes 3 and 4 are for instance nested on another property. In this case you can create new variables using for instance excel with an if-statement in which you say if your variable has option 3 or 4 it represents a one and zero otherwise. After this you can use binary logistic regression.