Hate to say it but provided you have a conventional old style auto(theres about 50 different types nowadays that have popped up in the past 5-10years), it sounds like a gubbed torque converter. Meaning a gearbox out job and most likely a new/refurbed/rebuilt unit. The price depends a lot on the car but with UK rates your looking at £500-£1500all in.
Fear not though it could be any number of things which are likely far cheaper
1. As others pointed out, dodgy or contaminated fuel, a mechanic could quickly and easly confirm this
2. Dodgy fuel injection, again easy to check, although this would be felt in all gears just more prodominantly in 1st
3. Gearbox fluid low/contaminated. Have this changed first. It may resolve the issue although possibly only temporarily if damage has already been done