Do I need to separate?
 

So I purchased an orchid a few months ago that has been doing really well. I noticed it has new growth and wondered do I need to separate and put it in a new pot and if so how is the best way to go about it without killing either of them. Sorry if this seems a daft question I am completely new when it comes to orchid care.

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No. That is a dendrobium and they like to be "pot bound" so there is room for several more canes in that pot.

Now, to the next point.... I hope that pot has sufficient drainage because it looks more like a fancy ceramic that may not or perhaps it's just a display pot with the real pot nested inside? Difficult to tell from pic. Just make sure when you water it, do it thoroughly and let it drain. Do not allow it to sit in water for anytime at all.

It is a Dendrobium nobile hybrid. Fertilize (light dose) regularly through Sept 15th, then stop.
Water well through Oct 15th, then cut back 90%.

It should then produce buds in late winter/early spring.

Once you see buds, start to increase watering.
Once buds open, resume full watering & fertilizing.

Good advice above.

Those plants (and many other orchids) live as colonies of individuals connected via a rhizome, through which they share nutrients, fuel and water. The bigger the colony, the more robust it will be.

The general rule of thumb is the minimum size should be 2 or 3 older growths plus a new one, but I think it’s best to let it get as large as you can comfortably deal with.

When well grown and flowered these are very rewarding plants. As I recall, they have the ability to rebloom from older canes (but not as profusely).

Welcome to the Orchid Board!

Look at this thread for growing information from a Japanese company specializing in this kind of Dendrobium:
Need Help with my new*Dendrobium Wave King Akebono

If the "inside pot" is one of those flimsy soft plastic things with one hole, then a repot would be in order. If you decide to do that, wait for new growth to start and then disturb the roots as little as possible (and certainly keep the plant together). You would choose small/medium bark to give good drainage You'd want a pot with good drainage (the ones with slits on the side are great, but plenty of bottom holes do the job too), about the same size.