Take a detour to Castle Morne in the south and beat the boss there to get the Grafted Blade Greatsword. It’s a great strength weapon all around and fantastic for the early game, especially since it’s easy to get.
I’d level up vigor and strength, though if you’re impatient you should level up strength first. The grafted greatsword has a STR requirement of 40, but you can two-hand it at 30 I think.
I’ve done two strength runs and each one I started the same way. Rush to get that sword, get 30 strength, dump everything else into vigor.