Can anyone tell me the main difference between these words and where they should and should not be used? Are they more or less synonyms?
All three have their established usages in a multitude of highly specific contexts (e.g. род "genus" vs. вид "species"), but in everyday usage, you often get situations where you can freely pick between at least two out of the three.
It helps to identify whether you're dealing with a set expression. Вроде "like, such as" is clearly its own word by now, even from its spelling, but equally immutable are
такого рода "suchlike, of this sort"
иного рода "of a different sort" (you also occasionally get такого/иного сорта);
в некотором роде "in a way" (rare, very bookish and often ironic)
что-то в этом роде "something of the sort"
в виде "in the form of"
типа "similar to, of the ___ kind"
Типа has also, since about the 1990s, become a very flexible "kinda"/"sorta" filler word that, like in colloquial English, can break syntax in very informal Russian: я типа понимаю "I get it and everything", а она типа: я уже знаю "and she's like: I know already".
But what if you just randomly need a word with that approximate meaning? To avoid being bogged down in existing idiomatics, let's take flying teapots as an example.
In the very rare event you'd want to say род летающих чайников, you'd mean it disparagingly. As in тот род летающих чайников, от которого все беды "the sort of flying teapots that are causing all the trouble".
And вид летающих чайников vs. тип летающих чайников is down to whether you're classifying by simple observation (вид), or some knowledge or assumption about the flying teapots' inner workings or mode of function (тип).
Здешний вид летающих чайников красивее. "The local variety of flying teapot is nicer-looking".
Только совсем другой тип летающих чайников смог бы поддерживать высоту в атмосфере Марса. "Only a wholly different type of flying teapot could maintain altitude in Mars's atmosphere."
Note that this (and basically anything regarding вид vs. тип unless there is clear linguistic precedent) is about which is a better fit, not which is correct. They are more or less synonyms, in that you have to try really hard to make it register as an error. The worst you can get is "mmm... okay". Even randomly using род in any of the last two examples would just be "mmmmmmmmmmmmmmm... o...kay?"