How do you weld carbon fibre to another piece of carbon fibre? If it can't be welded together what other solutions are there for connecting carbon fibre together?
You basically can't.
The same way carbon fiber is made: Epoxy. Fiber-filled if need be...but won't be as strong as a weld though.
The only other potential method I know of is to get carbon fiber composite that uses a thermoplastic matrix like nylon and try ultrasonic or friction welding it. But not only is nylon thermoplastic difficult to find raw, you need heated pressure molds to form it, and after all that I am not certain ultrasonic or friction welding would work in the presence of the carbon fibers. The welding itself also needs pressure molds.
And all of these methods have involve lap joints where fibers do not run across the joint, which is key of strength actually matters. A metal weld can be stronger than the surrounding material. Not so with composites for this reason.
The bonding in carbon is directional which means one carbon atom must bond in a fixed way to other atoms, e.g. 4 single bonds to hydrogen or two double bonds to oxygen. In the case of carbon-carbon bonding, you have one carbon to four others in diamond and one to three others in graphite.
The bonding in metals is delocalised which means that the metal atoms generally do not need to be ordered in a specific way to bond to each other. They have a preference to do that (hence they form crystals) but it is not a requirement.
So the point being that you can weld metals (with some exceptions) because their bonds are delocalised and they will happily join up with neighboring metal atoms; you can get a polycrystalline structure in a metal sample that is still a single cohesive piece. For graphite bonding to graphite - unless you can stitch at an atomic level these directional bonds, you can not "weld" or join two separate graphite parts.