You don't need any trading account to start a SIP in a mutual fund. Just get a form from the website of a mutual fund (Asset Management Company) like HDFC mutual fund (
http://www.hdfcfund.com/ ), fill up the form and submit it along with cheque and other documents to the address mentioned in the form and KIM. That is all you need to start a SIP.
Honestly speaking, you will need to have more money that 3K-5K for day trading. Though you can invest as little as Rs 100 in a SIP of a mutual fund.
You can buy few shares of companies you like and hold them in your trading account for long term. Say you can easily buy 3 shares of reliance industries every month for Rs. 3000 approx or about 10 shares of Hindustan unilever every month and hold those shares for long term. Open a trading account with say HDFC Securities along with a bank account with HDFC Bank. You can also open a trading account with sharekhan.com with a bank account with say HDFC Bank. Sharekhan has good customer support for newbie traders.
Remember, no amount is too small when it comes to savings. Small drops make a big a ocean. It is good to know that you are looking to start saving right at the very start of your career.
Happy journey.