My wife and I have a UK visa valid for 180 days. We will reside in the UK with my wife's sister (a British citizen). From the UK, we would like to make 3-4 weekend trips to various Schengen countries. Our trips will be sponsored by my wife's sister, who may be accompanying us. She will arrange our trips. How do we apply and get the visa?
I'm afraid most Schengen consulates in the UK won't routinely accept any application from someone on 180-day visa for the UK (standard visitor visa?). See Can we apply for a Schengen tourist visa while visiting another country on a short-stay visa? for more details on that.
Additionally, your wife's sister's sponsorship isn't worth much in this context. She isn't a resident in the Schengen area and therefore cannot provide an invitation or purpose for the trip nor can she promise to offer accommodation. She could fund the trip but you should still provide evidence on your financial situation to convince the consulate that you won't abscond. Furthermore, the family link is too remote to invoke your wife sister's EU freedom of movement (it only covers her spouse, minor children and in some cases parents, but not siblings, let alone their spouse).
You will therefore need to qualify for the visa on your own and ideally apply for it before leaving your country of usual residence (India?) using the regular procedure. If you are already in the UK, you can still try to get a visa but you should explain why you could not apply for it before.