Paralympic and world champion Simmonds took the 400m freestyle S6 title in five minutes 28.31 seconds, and Rodgers won the 400m freestyle S7 in 5:22.74.
Slater triumphed in the 100m backstroke S8 final with a time of 1:17.42.
Amy Marren won silver in the 50m freestyle S9, as did Andrew Mullenin the 100m freestyle S5.
Britons Jack Thomas and Aaron Moores won silver and bronze respectively as home favourite Marc Evers, 23, retained his 100m backstroke S14 title.
The Dutchman, who is also the Paralympic and world champion, clocked a world-record 1:00.44.
Recently crowned Commonwealth champion Ollie Hynd could not add the European 100m backstroke S8 title to his collection, coming fourth.
But there were eight GB medals on the opening day of competition with Jessica-Jane Applegate securing 100m backstroke S14 bronze, while Chloe Davies finished fourth.
Simmonds, 19, was “really happy” to win gold, but said: “It was a rubbish swim. I’m not happy with the time.”
Slater took inspiration from her S8 100m freestyle silver medal at the Commonwealth Games.
“It was a two second PB tonight after already knocking a second off this morning,” she said. “I didn’t expect to go that fast to be honest as it’s not a main stroke of mine.”
Rodgers said: “It was one of those ones where you don’t know what is going to happen.
“I had a race plan and I’m pleased with how it went. It was a good time. We are going in the right direction.”
The British team also features London 2012 Paralympic champions Josef Craig and Sascha Kindred – with all the para-swimmers competing in multiple events over the six days.
Spain’s six-time Paralympic champion Teresa Perales made a winning comeback from illness, beating Norway’s seven-time world gold medallist Sarah Louise Rung in the women’s 100m freestyle S5, clocking 1:20.32.
There were also records tumbling with Ukranian Oleksii Fedyna setting a new men’s 100m breaststroke SB12 world record and Russian teenager Alexander Makarov breaking the 100m freestyle S2 world record.
The championships run until Sunday, 10 August.