I got a L size Opus Allegro 4.0 (2014) which came with Vittoria Zaffiro tyres 23-622 (700*23c) on an Alloy Shimano Asoo WH-R501 wheelset 622x15c. I'd like to fit a wider tyre to improve my experience and feel when I am on bumpy roads.
I'd like to know what parameters I should consider when I fit a wider tyre (say a 24 or 25) on the current wheels/innertubes and, what are the limitations of my current set up if I don't want to change all at the same time (i.e., bike frame, etc.)
The two things to consider when increasing tire size are compatibility with the rim width and frame/fork clearance. Inner tubes fit a range of tire sizes.
15mm rims will easily take 23 or 25mm tires. Check that your frame has sufficient clearance for an extra mm or two on either side (some tires are a bit wider than specified width) at the brake calipers, fork crown and chain stays. I'd be very surprised if the frame does not have clearance for 25mm tires.
23 to 25 is pretty minimal - you might not feel much of a difference.
I'd look at the chainstay and seatstay gap, and then the brake bridge to see how much space you have, and then go for the largest tyre that will fit without causing frame rub.
Different tyre brands have subtly different sizings, and they vary across rim widths too.
Also remember frames flex while riding, so that which fits statically may cause frame rub in a dynamic ridden environment.
Your question about tubes is answered elsewhere on the site - start with:
Will a "700 x 25-32cc" inner tube fit my "700Cx28/38c" / "ETRTO 622x20" rim? and read the linked questions etc.
If your inner tubes are 18-23mm, then by all means go for new tubes. However, current smallest tubes from both Conti and Schwalbe are 18-25 compatible, so you should probably be fine.
