Start with the obvious one, because it catches more people than you'd think: is the fridge actually running? Open the door and listen for the compressor and the fan. Silence with a light that still comes on usually means a control or start-relay problem. Silence with no light means you're chasing a power issue — check the outlet with a lamp before anything else.
Next, look at the freezer. If the freezer is cold but the fresh-food side is warm, the sealed system is almost certainly fine. On most refrigerators the fresh-food compartment is cooled by air pushed over from the freezer, so a failed evaporator fan or a defrost problem stops that airflow while the freezer itself stays frozen. That's a repair worth making — the expensive part is still healthy.
Third, pull the freezer's back panel if you can reach the screws and look at the evaporator coil. A solid block of frost around it means the defrost cycle has stopped working: a heater, a thermostat, or a timer, none of which are catastrophic. Defrost it fully before you decide anything, because a coil buried in ice can't cool no matter how well the compressor is running.
Fourth, check airflow around and inside the unit. Vents blocked by a full shelf, condenser coils packed with dust underneath, or three inches of clearance behind a unit that needs six will all cause slow warming that looks like a failure. Vacuum the coils and give it a day before you call anyone.
Fifth, listen for a compressor that clicks and hums for a few seconds then stops. That's usually a start relay or an overload protector, and it's one of the cheaper fixes on a refrigerator. If none of these describe your fridge, that's genuinely useful information to hand a technician — it narrows the diagnosis before we've walked through the door.