Start with the VIN plate. The first weight is the car’s MAM (gross vehicle weight), the second is the gross train weight, then the axle limits. Towing limits and kerb weight are in the handbook. The trailer’s MAM is on its own plate.

The car

FigureWhere to look
MAM (gross vehicle weight)VIN plate, weight 1. On a registration certificate in the EU format (the V5C and the Irish certificate) it is field F.1 (Directive 1999/37).
Gross train weightVIN plate, weight 2.
Towing limit, braked and unbrakedHandbook or specification sheet. The registration format has fields O.1 (braked) and O.2 (unbraked), but they are optional, so not every certificate shows them.
Kerb weightHandbook. Field G on the registration certificate is the mass in service, which can differ from the handbook kerb weight. The GB legal definition of kerbside weight leaves out the driver.
Maximum noseweightHandbook, or the tow bar’s own label.

GOV.UK's plate example lists weight 1 as the maximum weight of the car on its own and weight 2 as the maximum weight of the car and trailer combined (GOV.UK).

The trailer

The trailer's MAM (Irish documents say DGVW, design gross vehicle weight) is on its plate, usually on the drawbar. In Ireland, the RSA says all new light trailers sold after 29 October 2012 must be plated (RSA).

What it actually weighs

Plates give maximums. For actual weights, use a public weighbridge: weigh the car alone, then car and trailer, then the trailer on its own.

Questions

Is the towing limit on the V5C?
Sometimes. Fields O.1 and O.2 are optional in the EU registration format, so check your own document and the handbook.
What is MAM?
Maximum authorised mass: the most the vehicle or trailer may weigh fully loaded, as plated. Licence rules use it.

Sources

Guidance, not legal advice. Towsheet is independent and not connected with the DVLA, DVA, RSA or NDLS.