I have a Triban RC500 bike. Apparently it's wheels can take wider tyres.
A lot of internet info suggests that this is a good thing to do for comfort and perhaps for better rolling resistance too?
My questions is, will I need to change to different inner tubes or the current tubes are OK to go from 28mm to 32/35 mm tyres?
Common inner tube tire width ranges for ETRTO 622 rims are: 23-25, 28-32, 35-43. It's highly likely that you have tubes that will work with 32mm tires, and 35mm at a (literal) stretch.
Don't expect going to wider 32 or 35mm tires will give you better rolling resistance. Wider tire at lower pressures generally have higher rolling resistance (on smooth surfaces at least, on gravel or bumpy surfaces different factors come into play).
The rims may well be capable of taking a wider tires but the limiting factor is the fork and frame, especially between the chain stays behind the bottom bracket. See What is the maximum or minimum tire width I can fit on my bicycle
Inner tubes will stretch to fit a bigger tyre, so it is likely to be ok. There is some risk that the stretching will make the tube wall a little thinner, and it can be more prone to punctures, or it can expose a weakness and cause a flat.
Personally I would change the tubes at the same time, as tubes are cheap. You could likely keep the old tubes with the old tyres in case you wanted to go back to the smaller tyres.
Another issue may be clearance of the frame. The bike might not accomodate the wider tyre. Some times the rear wheel will not go into the dropouts properly when the tyre is inflated. That is not fatal, just perhaps annoying that the tyre may need to be deflated when putting the wheel in. Makes it difficult when fixing flats out on the road.
