you have not mentioned who is hindu and muslim
one way and immediate marriage is both should be hindus or the other religion person has to convert to hindu and get married in Arya samaj and both should be majors means boy 21+ and girl 18+
in case of special marriage act no religion it is immaterlal
Court marriage as you call it is solemnized under the Special Marriage Act, 1954 all such marriages can take place between any two Indian citizens as well one Indian citizen with a foreigner irrespective of their religion, caste or creed. Such a marriage is solemnized before the marriage officer/registrar of the area (Sub registrar/SDM is the marriage officer/registrar only in case the already solemnized marriage under any other Act has to be registered) where both or any of the parties to such a marriage reside permanently or for at least one month. Cost of such a marriage is merely Rs. 90/- or as the rate prevalent in your area.
Both the parties to the marriage file a joint application before the marriage officer along with individual affidavit attested by notary public specifying the age(boy 21yrs &girl 18yrs), place of residence, marital status(unmarried or divorcee as the case maybe), mental status(not idiot or not with unfit mind), not related to each other. Along with this you have to give proof of age , proof of residence(Passport, ration card, voters id card, PAN card, rent agreement, even in your case hostel certificate stated period of your stay) & the application of such a marriage should be witnessed by 3 independent adult persons, affix one photograph of each of the parties & all the 3 witnesses. You have made such an application/notice to the marriage officer in person along with all the 3 witnesses. On receiving such an application the marriage officer issues a month notice regarding the intended marriage & get such a notice affixed on the notice board out side his office, a copy of such notice is sent to the marriage officer under whose jurisdiction the other party lives, also a copy is to the individual parties address. After expiry of the period of notice a date is fixed on which all the persons parties to the marriage & 3 witnesses appear again before the marriage officer who declare them married & issue a certificate