I think it's impossible to avoid hills given that Horniman hill is between them! You'd have to go a very long way around to avoid it.

That stretch of London Road (from Sainsburys to the Grove) isn't too bad to cycle (despite it being a hill) since there is a lovely big bus lane on the actual hill so you don't end up squeezed by traffic. And while it will be tough for the first week or so, the best way of getting better at going uphill on a bike, is to go uphill on a bike! Just use your gears - and challenge yourself to keep going a little further each time.

The pavement along Dulwich Common is shared use so you can then hop onto the pavement (legally and considerately) along to the Park. Or you could go straight on at the lights and turn down Court Lane and get to the park that way.

The only other option I can think of is going onto Honor Oak Road (which is narrower than South Circ so feels worse to cycle along) and trying to get over to Wood Vale so you can go down Melford Road and up to Court Lane that way. But there's a steeper hill involved there - and it seems a roundabout way for a short cut.