1)I have five lakhs in hand.Please suggest me some funds to invest this lump some amount of money at once.

I would recommend not to invest this all at a time.Instead invest in small amounts every month in the form of SIPs. Here are some good funds which are performing currently well in the market.

Diversified Equity : DSP BR TOP 100, HDFC TOP 200, BSL Frontline Equity (Plan A).
Debt Funds : Fortes flexi debt fund , canara robecco income funds.
Tax saving funds : BSL Tax Relief '96, SBI Magnum Tax gain, Canara Robecco Tax saver.

Any time don't go for more than 6 funds. Your investment ratio should 80:20 for diversified and debt funds respectively.

2)I want to invest 3000/- rupees every month towards mutual funds.Can you suggest how to start with?

It depends on how much risk you can bear.If you want to take high risk,then you can go for mid cap funds like Reliance Growth Fund, Large cap funds like DSP BR TOP 100 or HDFC TOP 200. Incase of thematic funds you can for DSP BR Tiger fund or ICICI Prudential infrastructure fund.