I have 14 AWG wires that need to connect to the + and - of the below PCB board. What way should I connect it to provide the cleanest look? Is there some sort of connector I can attach the wire to first, then solder to those tabs?
Those tabs are designed to have wires soldered directly to them. Trim the insulation so that it comes to the end of the board and the wire overlaps the entire pad. Pre-tin the pad and the wire and just melt the wire into place.
The two circular holes at the one end are for the XT60 connector which you will find on many LiPo batteries.
As this appears to be a voltage regulator powered by a LiPo battery, you should ensure that whatever connector you use prevents you from accidentally shorting the battery cables. For this reason i would avoid the screw terminals. It also needs to prevent you from accidentally reversing polarity. The XT60 connector in the picture ensures both, so my 1st thought would be to use the same connector and an extension cable. Since the vias are plated I believe this is not too hard and it just might fit. If it does, read ahead. Otherwise you can skip the rest of my answer.
To help the solder fill the gaps, clean the via (hole) and the connector with something like isopropyl alcohol, check you have the correct polarity (the connector it the right way around so when the battery connects + goes to + and - to -). Apply some solder on both surfaces first and then solder the connector directly on the PCB. This last step might be a bit tricky if the inside of the XT60 connector legs fit too close to the sides of the vias in which case I would only tin the outer ends of the vias slightly - just to make better thermal contact when I touch the soldering iron.
If the gap between the connector legs and the via is to large, you could try filling part of it with some copper wire before soldering.