The lights dimming are a sign of high (excessive) current draw. Some where a wire is heating up.
What is the wire run length from you breaker panel to the garage? that would be down from the panel to the ground, across the distance (25' ditch?) and up to where ever you wanted a plug.
A "dedicated circuit" is one plug.
For $800 I'd do it myself
A ditch witch can be rented and cut a 25' slit trench in less than two hours (way less). If the run is not too far #10-2 w/ground, direct bury wire could be used though #8 would be better.
At the breaker box one additional 220v 20 amp ckt. breaker would need to be added, an exit hole found and a wire clamp in the hole. If the panel is in a wall the wire would have to be fished through or cut into the wall and repair later.
If the wire exits the house outside, a short piece of plastic conduit would be needed to get below grade and likewise up at the garage. One 220v plug and box to install it into nailed or screwed to a wall or stud.
One light circuit per leg. One light switch ($ ~0.60) one more box. If lights are to be individually switched , two light switches another box or a bigger box.
I'd add a ground rod at the garage but I'm a nut on grounds.
Home Depot/Lowes sells a little book that explains wiring.
I am not an electrician.