The whole “you need an amount of energy approaching infinity to reach the speed of light with physical matter” principle is sound. It's just not that hard to get infinite energy. Build a energy amplification engine or perpetual motion machine, hook it up the ship, and get blasting.
 Speed of Light doesn't work the way we think it does. Just put on more rockets and boosters. It will go faster then light, don't worry about it.
 The Speed of Light being a constant in the universe is actually just a time dilation problem in appearance. Away from any planet or gravity well, relative time slows to a crawl. You can reach distant stars of the galaxy in few months travel objectively, the time of travel in between the gravity wells just stretches on and on. You need immortal computers or cryogenic stasis to get crew there.
 Do the whole wrinkle in time thing. Artificially bridge the distance between two places much shorter then it actually is; bending space is easy. Can also be used as a weapon; entire spaceships are compacted into the space of a micrometer if directly in front of a wrinkle-ray.
 Hyperwaves. Just use some yet misunderstood type of energy to shift the ship into a hyper dimension. Here, a few hours of travel equates to hundreds of millions of light years in normal space.
 Literal magic. Scientists got tired of trying to figure out the blips of unknown knowledge. Started to wave sticks at it and say mumbojumbo until it worked. Eventually a spell that can help things move faster then light was discovered. Nobody knows how it works.
 Psychic powers. Ships can travel faster then light based on using the “speed of the mind”, which is faster then light apparently.
 You go through a subspace dimension, where distances are closer. The subspace dimension can have evil warp demons, dimensional god-beings, or even more strange alien races living in the “spaces between spaces”.
 Direct current of electricity has a very special property. Gravity along the “line” shifts towards the “end” the more powerful and long lasting the current is. In other words, after some warm up, it makes things go really, really fast.
 Transportation across vast distances is possible with quantum entanglement. However, the type of software and advanced scanners needed to detect and anchor to these points are heavily reliant on gravity, and aren't super precise. This means you can warp to gas giants, but are reliant on local transport to get around the smaller moons and planetoids. Derelict ships in deep space are totally stranded and shit out of luck.
 Yet unknowable cosmic entities only allow FTL travel with the sacrifice of a cute animal pet. It's sadistic as hell and yet that's the only way they know how. It can't just be a stray either, it has to be cute and loved by at least one person. Assholes.