Rivers, Water of Leith, which flows through the town is free but does require a permit and has some decent trout. Peebles association water, 23 miles of river and less than an hours drive from the city. Beautiful location and possibility of specimen trout. River Clyde about an hour drive from the city and again, the possibility of specimen fish. Both will cost around a tenner for the day. There are a few others in east and west Lothian which are possibly nearer and cheaper.
Lochs, Harperrig 30 to 40 minutes from the city bus at £15 is grossly overpriced for fishing for what are mostly half pounders. More than that for a boat. Gladhouse reservoir is lovely location with better fish but unsure about costs and boat availability.
Whiteadder resevoir is bank fishing about 1 hour 20 minutes south east of Edinburgh and will cost zero because the ticket machine has been broken for years now. Half pounders again though. Others will give more suggestions. Hope you enjoy your visit.
The water of leith season finishes on the 30th of September, but plenty of small wild fish to be had, dry flies work well (small griffiths gnats, grey dusters and Adams patterns are good). Hareperrig is full of wee wild fish but is quite expensive for what they are. Threipmuir reservoir by Balerno is stocked With browns twice a season and has a head of wild browns (No rainbows), permits can be had online from malleny angling Ltd, my fishing club, good quality fishing to be had and if the browns are playing hardball the other reservoir (Harlaw) can be fished off the same permit. Harlaw is stocked with rainbows though.
The various border reservoirs mentioned are good, the better river fishing is further afield. The tweed and a few of its tribs are mostly around an hours drive away, the Clyde also. A lot of central Scotland can be Reached within an hour or two drive and there is a multitude of options for fishing.
What type of fishing to you want specifically and how far do you want to go.