First, congratulations! You'll soon find out that there are only two kinds of car drivers: those who drive Citroëns and those nincompoops who don't... :-)
Start with a major cleanup of all compartments: clean the carpets, under the front and rear seats, in the trunk. Clean the exterior, the wheels. Check the car body thoroughly for scratches, small dents and rust. Rust should not be a problem on a BX unless it was neglected or some repairs were carried out without paying attention to prevent rust formation. Repair as necessary.
If the car is equipped with a catalytic converter, take care of the general precautions: you should avoid all situations where harmful substances could reach the converter. To summarize: to avoid unburned fuel overheating and destroying the converter, do not push or tow start, do not use the car with too rich a mixture (have the air cleaner, fuel filter, spark plugs cleaned and renewed as required), do not allow the engine to misfire, do not switch off the ignition at high speeds (in particular, do not blip the throttle before switching off). To avoid unburned carbon deposits, do not allow the engine to consume engine oil excessively. To avoid other harmful substances, do not use fuel or engine oil additives. To avoid lead, use only unleaded fuel. Also note that the converter becomes very hot in service. Do not park the car over long grass, piles of dead leaves or other combustible materials, especially as it will lower itself when the suspension is not under pressure.
Replace (or have it replaced) the camshaft belt, unless you know the previous owner personally and can trust him/her if he/she says that it had been done recently. The belt is relatively cheap but the damage it can cause, should it fail in service, can be catastrophic (the least damage is a broken camshaft but it can destroy the whole engine easily).
Also, replace the radiator filler cap. It is cheap and its spring loses its strength over time, leading to engine overheating and possible engine damage. If your engine overheats, even if it's only the pre-warning lamp (yellow or flashing light, or temperature between 95-100 °C) take it very seriously and track down the failure. These engines are not supposed to reach this temperature, not even sitting in a traffic jam in a hot afternoon, hence, every overheating signals a failure that should be paid attention to. These problems, if found and rectified early, are relatively cheap, but blown head gaskets, warped engine heads and other major failures caused by repeated or prolonged overheating, are definitely not.
Replace some (or most) of the liquids. The engine oil should be changed immediately. I would suggest fully synthetic oil, the price difference today, although still present, is not that huge, and its broader temperature range and stable viscosity help keep your engine in good condition. Of course, if the engine is consuming too much oil, this wouldn't be an economic solution, but excessive oil consumption is a failure that should be repaired anyway. If you had no previous experience with Citroëns, you might be surprised, but these engines, if they are in good condition, practically do not consume—or only a very small quantity of—oil between regular oil changes (every 10,000 km by factory recommendation, every 7,000-8,000 km if you want to be perfectly sure and don't mind the additional expense, every 5,000 km would already be a little paranoid). Check your oil consumption carefully in the first period to see if there is some work to do on the engine.
There is one thing of utmost importance when repairing any hydropneumatically suspended Citroën: never do anything under the car, especially not any work on the suspension, without the car safely supported on axle stands, lifted with a car lift or standing over an inspection pit. Even over an inspection pit, always allow the extra distance for the body to lower itself. Never rely on the height lever set to highest setting or only on the jack provided with the car to change tires. Don't let anybody sit into the car while you work under it (eg. by locking the doors). If the car drops abruptly, it can crush anybody to death in a second. This is not over-anxiety: it has already happened. Don't let it happen to you!
The coolant should have been drained and renewed every second year. However, most owners never do this. Hence, it would be a reasonable decision to flush the radiator and renew the coolant as well. But even if you decide against it, have the freezing point checked to see its condition—many garages or gas stations check this for a small fee or even free, or you can buy a simple meter at general car accessory stores to check it for yourself.
When you have the coolant drained, do not forget to check the cooling fan temperature switch (screwed into the radiator, bottom right) whether it makes and brakes at the specified temperatures. Also, a cheap part, easy to replace, and tends to go wrong over time (especially when the low speed resistor on 16 and 19 engines—an aluminum cylinder right behind the left headlamp—is missing or failing) causing engine overheating.
If you have the service records and they show that the transmission oil is not yet near the end of its expected lifetime, you can leave it alone. If you have to change it, buy mineral oil. Synthetics are perfect for the engine but are a bit too thin for the gearbox of the BX, resulting in less smooth a gearchange. The LHM (the hydraulics liquid) deserves some attention, too. Start by cleaning the filter in the reservoir. If the LHM already looks yellowish instead of bright light green or if you want to be sure (and especially if you found any greater amount of dirt in the filter), flush the system. Do this before anything else if you suspect that the ride comfort is inferior to what can be expected from a BX (unless you found an obvious failure in one of the suspension components).
Spark plugs and air filter are also natural candidates for renewal (but clean them at least). Clean the distributor cap and rotor. Clean the connectors of the ignition coil (on most BXs the coil is located right behind the battery; the fumes emitted by conventional batteries often corrode its connectors). Check the tires for thread wear. Check the condition of the brake pads and disks. Although the front pads have a warning lamp in the instrument panel, don't count on it—after a few years, the switch will probably be out of order. Check the rubber bellows on both ends of the driveshafts: if they start to crack, they can be replaced for peanuts. If they stay open for a while and allow the grease to escape and dirt to enter, it will be much more expensive.
Examine the engine and transmission carefully for oil and coolant leakage. If you find none, clean the engine and transmission (if everything is perfectly clear, you might suspect that the previous owner wanted to conceal something leaking. But then it might mean that he or she just liked the car to be clean...). Check all rubber hoses, gaskets, seals and belts, tighten the retaining clips of hoses if necessary. Drive the car for a week or so and carefully check for leaks at hoses and gaskets.
Remove the lens of the rear combination lights. Clean the lens from the inside, the bulbs and the reflector surfaces with soapy water. Renew the bulbs if they are already discolored.
If you still have patience, go methodically through every electrical connector in the car, disconnect, clean, treat with a contact cleaner (not WD-40!) and reconnect. Don't forget to clean the grounding points as well. This can help avoid many annoying problems in the future.
And, in the future, pay attention to the activities described in the regular maintenance table.