Once you have cleared whatever type of trap you have been working on and replaced it, you don’t have to over tighten the securing nuts as this may distort the rubber seals.
Its usually best to tighten moderately and then run the water to observe how well the trap is sealed.
If you witness any drips, you can tighten the nut further, but not excessively, to avoid distorting the seals as mentioned.
If you still have a leak, the trap may well need new seals, these can be bought from any plumbers merchants quite cheaply.
The pipework for a wash hand basin is typically installed in 32mm pipework and a kitchen sink in 40mm pipework, the seals and traps for these being the same size.
If in doubt, a new trap, complete with seals can be bought at the same merchants quite cheaply too and this option may just make it a little less complicated if your worried about fitting new seals and the various sizes, but if you take the old trap to the merchants, they will match the seals or trap for you, taking some of the guesswork out of it all for you.
If after all of this, the sink or basin still doesn’t run properly, the problem is probably more deep rooted and require us to clear it with any number of tools at our disposal, but its always worth trying to clear the trap in the first instance as if this does the trick, its cost you very little and added a new skill to you DIY repertoire.
Good luck, and remember, we’re always here to advise and support you and if the worst happens, clear you drainage problems for you.