This answer is based on another answer of mine (which I've also adapted here), but this version is adapted to your question slightly.
If you're not sure your items are all measuring the same latent construct, you probably should consider exploratory item factor analysis. See these questions for some tips:
If (or once) you're sure your six items are all measuring the same latent construct, you could use a partial credit model to account for differences in response scaling across all six items. If the four items with 4-point Likert scale (polytomous) measurements are all on the exact same scale though, you might be better off dropping the binary items and using a rating scale model of the four polytomous items, depending partly on how much unique and valid information you get out of those binary items. John Michael Linacre and Benjamin D. Wright posted some discussions of the differences between partial credit and rating scale models over at rasch.org that might give you a better sense of what you'd be dealing with if you go the item response theory route here.
Some latent variable analysis programs will let you set certain thresholds to be equal across certain items and leave another item's threshold freely estimated. You might be able to blend the partial credit and rating scale models this way by setting your six polytomous items' thresholds (each item will have three) to be equal across items, and estimating the binary items' single threshold independently (each could also have its own threshold, depending on how different the binary items are)...but I'm not exactly sure this is all you'd need to do to have the best of both worlds.
The simple, "classical test theory" approach that weighs every item equally would probably have you just standardize all the items and average the $z$-scores, but I don't think that's a good idea, because four-point Likert scales may not approximate a continuous dimension well enough (and a binary item definitely won't; it might not even make sense), nor can you be sure that the average of four polytomous items will be approximately continuous enough. I've seen it suggested that each item's Likert scale should have at least five options to approximate a normal distribution, and at least five Likert scale items should measure the same construct if their simple sum / average is to approximate a continuous dimension. (I've seen Berry, 1993 cited as a source for an argument that five or fewer options is unacceptable, but I haven't read it myself.)
Berry, W. D. (1993). Understanding regression assumptions. Newbury Park, CA: Sage.