Hi and welcome to the forum OP.
Great choice by the way - I'm now on my second, as they are a bit addictive!
There is plenty of good information on here if you search the forums, but hopefully you already know the Coupe only came with the 3.0Si N52 engine (or the 3.2 litre M engine, but that is a Z4M). So at least that makes choosing which one a bit easier!
The N52 is a great engine, and if properly looked after seems to be capable of high mileages (someone on here was selling one a while ago with over 200K miles on it). They don't have a dipstick so don't bother looking for one! You check it by pressing a button. They have an electric water pump which seems to fail anywhere around the age of 8 to 10 years, which all of them are now! The pump in my first one went at 9 years and 60K miles. Euro Car Parts quoted £340 just for the pump but I've got nowhere to do DIY jobs so I had my local BMW Indy change it and he used an OEM BMW pump which was £500 plus fitting!
So if you look at a car that has already had one that is a bonus IMHO! The only other mechanical issues I can think of are ABS pumps and Electric Power Steering Pumps, but I believe they can be repaired fortunately, as new ones aren't cheap! But I think issues with those are pretty rare compared to the water pump.
Like most BMWs of the era they are also prone to back springs breaking, but they aren't dear!
But don't be put off - they are great looking cars that drive so well they always put a smile on your face!
Check it has good service history and has been looked after - plus buying privately you get to meet the owner which can help you form an opinion! Check service history - it needs condition based servicing so usually every 16,000 miles and no more than 2 years, whichever comes first. The mileage left until a service flashes up briefly on the dash when you start it up. But a lot of members on here do an interim oil and filter change.
They all came with Bridgestone RunFlat Tyres, but most on here think a swap to non-RunFlats helps with ride and handling. Especially as they are prone to tramlining - most reckon ditching runflats, changing any worn bushes and a 4 wheel alignment will fix that. Worht checking all tyres are the same - I looked at one with 2 runflats and 2 non-runflats which wasn't ideal!
They came in SE or Sport spec. Sport has M-Sport suspension (stiffer and 10mm lower) but more importantly M-Sport seats which are just so much better!
Colour choice is limited, especially if you don't care for silvers or blacks!
And standard spec is pretty basic! They didn't come with rear parking sensors, cup-holders, cruise control or a Multi-Function Steering Wheel; heated seats, Xenon headlights and higher spec stereos are also desirable options, so you need to decide what is essential and what you could live without as most Coupes don't have all the options!
Lastly my advice would be to be prepared to travel! Based on figures another Forum member got less than 2,500 Coupes came to the UK.
Good luck finding one you like, and when you get it don't forget we like photos on here!
Because of that, and maybe by way of inspiration here is my current one:-