Should I use the ‘quick apply’ buttons or go to the company website and apply?

It is always best to view the advertisement directly on the site to check the details are accurate. You can then be sure that the ad is genuine and worth spending time on the application. If you cannot find it on the site, consider reaching out to the organisation  to check or failing that, it is never a bad thing to just apply through ‘quick apply’ if you have exhausted all other options. 

Should I bother applying if I’m missing some of the experience in the advert, but I meet most other criteria?

Absolutely! Organisations will try to recruit the ‘perfect’ candidate by asking for every single skill required, who usually does not exist… if you feel you could competently do the role (with maybe some development at times) then it is always worth applying and being prepared to state at interview how you would enhance your lack of skill/experience in that particular aspect.