Starting 340 Build w/ Supercharger

Sure you can blower that 340. If it means making your dream come true. But you're gonna need to spend some money on the chassis to get it to hook, cuz if it don't hook, you might be big-time disappointed. Cuz then it might as well be a teener.
If you just want a cool looking ride and don't care about pulling the front tires, then, yeah a blown 340 is really,really, cool. That 340 can be an instant 400 to 450 horsepower, and it won't need to rev very high. But that's at 100% throttle. A streeter probably spends most of it's life down at 30 to 50% throttle, so like 200hp is plenty enough to just tool around with, and have a pretty good time. And you can never use 100% for more than about 5 or 6 seconds, cuz then you're ripe to get a ticket. If you get too many of those up here, they take your driving privileges away. So the way I look at is like this; 200hp is barely enough. 450 is really too much. Up the middle is 325, and that's about standard for a factory original hi-compression 340 with headers. Make it a 360 with the same parts and add 20hp, so 350 is easy. Now the tires are a lot happier, and they last a lot longer, and you're not down every week with a busted this or that.
To help get traction; the usual recipe is; relocated SS springs,long drag shocks,a narrowed rear,at least 10 inches of radial per side,a lightened and loose front end, and frame connectors are always a good idea. Some guys are Ok with running around on city streets with this recipe. ..... But it ain't fun in the corners. In a small town, the streets are maybe 4 blocks long, so there's a lot of turning involved.And for every second I goose it, I need 2 or more to slow it down! Ao never forget about the shut-down area. Never!
It sounds like you've had a driver's license for quite a while, so I'm sure you've turned a corner a time or two,lol. And so, you probably already know what you like and what you don't. Personally, I like stuffing it into a turn hot and bringing it around with the gas-pedal. So you're not gonna be doing that with a loose front and a sky-jacked rear.
But if your style does not include wreckless behavior like mine,lol, then set it up as you wish.Just be aware that 450hp is incredibly powerful, and dangerous, and needs a lotta brakes to get her slowed after 7 seconds at 100%. Oh, and the brakes don't work if the tires aren't on the pavement, or when you're sliding sideways,lol.And if it looks like you're about to end up going backwards, clutch it man!