BRAKES:
Remember that the most recent MGB came off of the assembly line over 25 years ago. If the rubber parts in the hydraulic systems have not yet been replaced, they will need it. Look at the fluid in both clutch and brake reservoirs. It should be clear or have a pale yellowish tinge (silicone fluid is OK, too. Its purple.) If it is foggy, cloudy or dark, it has not been changed in quite a while. It will need to be replaced and all the cylinders bled until the fluid comes out clean.
If they have not been replaced very recently, plan on three new flex lines for the brakes, as well as a clutch hose. Similarly, the master cylinders will likely need to be rebuilt or replaced. If you are lucky and the old fluid was not full of water, the cylinders will only need to be disassembled, quickly honed and a rebuild kit installed. However, the bores generally have some amount of pitting, and you are faced with the choice of either having your cylinders professionally sleeved and rebuilt, or replacing them with new ones.
Front calipers are very tough, and unless they have been parked for so long that they are completely frozen, they will usually work. Brake pads are easy to change, but you should have the rotors turned by a machine shop at the same time pads are renewed, whether or not they are replaced. Re-pack and set up the front hub bearings at the same time.
Rear cylinders are very inexpensive, and if they do not work smoothly or show any signs of leakage, it is more expedient to simply replace them. Brake shoes for the rear are inexpensive as well, but the drums need to be turned when replacing shoes. Inspect behind the hubs for evidence of leaking axle seals. If there is oil back there, the axle seals will need to be renewed. If this is left unchecked, the brakes will become oil-soaked and dangerous.
CLUTCH:
MGBs are fitted with a hydraulic clutch system, which offers the advantage of being mostly self-adjusting and idiot-proof. The disadvantage is higher cost when it comes around to repair the system, since there are more components involved.
The first evidence of problems with the clutch circuit (the hydraulic part, at least) is usually fluid leaking either out of the slave, leaving a puddle on the floor under the car, or the master, allowing fluid to run down the pedal and drip onto the floor mats or carpets.
The flexible hose leading to the slave can also fail, either externally, causing a spectacular leak, but also internally. As the hose degrades, a torn flap of rubber inside the hose can act like a valve, causing bizarre pedal behavior.
The master is removable from inside the car, via a rubber plug. Depending upon its internal condition, they are sometimes rebuildable, so long as there is no pitting in the bore. If not, fit a new one, they are not expensive enough to warrant sleeving.
The slave is usually a goner by the time they leak badly enough to warrant examination. For every ten I tear apart, maybe 2 are savable. Its worth having a look, though, since the kit is ridiculously cheap. If it is a goner, reproduction cylinders are of wonderful quality and half the price of original-branded ones.
Just replace the hose, its cheap.
One final note. Bleeding clutches can be a pain, since the hard lines tend to route higher than the master cylinder. This makes gravity your enemy. Once a bubble gets into that line, it can be a bear to flush out. Having two people to the pump and hold routine does not work quickly enough to keep ahead of that bubble. The best way I have found is with a pressure bleeder, either of the Easi-Bleed variety or a more robust unit built for repair shops. I have mixed success with vacuum bleeders.