Great question. For the sake of keeping my answer shortish, I am going to assume that you understand everything in the sentence before the comma. I'll edit the answer if that isn't the case. :)
There are two parts to that monster particle of yours: に and でも。 The に particle is the particle that indicates direction, someplace a person goes. In this case, に modifies 旅行。 So the first part of your particle simply means "on vacation"
でも in this case is a way of saying "... or something" I imagine that you have also encountered でも in your studies, and that it meant "but," or "however." These translations are correct as well. でも is actually quite versatile, and I suggest you look at the variety of definitions here.
Long story short, your sentence means:
You shouldn't put all of your effort into working all the time, you should take the time to go on vacation or something.
The bold parts are what にでも would appear in English.
たまには is another one of those weird Japanese quirks. You can actually remove the は from the sentence and still have it mean the same thing. This really hits on the meaning of the particle は。 The best way to put it in this case is that the は is putting an emphasis on the たまに in this sentence, in effect marking it as a pseudo subject. This kind of emphasis is typically implied in English, but in Japanese we say it. は in this case actually changes occasionally (たまに) to something more along the lines of "sometimes", "once in a while", or "from time to time". Be careful though, because not all に adverbs can be modified this way, but it's not unheard of either.
Note: in the first half of this sentence we encounter the ～ないで grammar that you asked about in a previous question. In this case it means "don't."