Adding disc brakes to the back isn't a cost effective swap for now and unnecessary unless you are racing or just have to show off that you have rear discs. Upgrading the front brakes will make a much bigger difference as they do most of the stopping. My car is pretty serious, I autocross it a lot and drag race on occasion besides drive it daily. I still have stock G body drum brakes out back and won;t switch to discs until I get around to installing a 9" Ford, which the brake swap will simply be convenient. I did upgrade to 98-02 Camaro front brakes and that made a HUGE difference. Upgrading to or Wilwood or Baer stuff would be nice but not budget friendly for you right now.
When I first bought my wagon I left the 305 in it for about a year or so and then swapped in a 406 SBC. It took a little swallowing my pride to drive a slow car but I didn't really care, it was super reliable and got decent gas mileage, and I knew more power was coming soon.
If you plan to run an LS I wouldn't bother wasting any cash on the 305 or a 350 in the mean time, save it for the LS. My son who is near your age is driving around in the stock 350 in his wagon, it's not real fast, mid 16s in the 1/4, but it's enough power to have fun autocrossing. The only upgrades are the cat fell off and it has a Flowmaster muffler to sound good. He bought a wrecked 2003 Tahoe from Copart for $800+ $300 in fees and $450 in shipping. So he got a VERY good running complete truck with a 5.3 and 4L60E for just under $1600. Doing it this way means he gets all the wiring, sensors, accessories like the alternator and power steering pump. He can part out the good parts on the truck and scrap what's left to recover some money. He bought a used set of 98-02 Camaro headers for cheap too. We will use the factory wiring harness and spend the couple hours it takes to do the pin out ourselves. A flash of the computer I understand runs about $175 and a friend has HP tuners so we can get into it. Ultimately we figure about $1000 + to complete the swap. I think a $2500 budget for an LS swap (without trans) is reasonable. You can save some $ by buying a complete vehicle.
4.8s might be more plentiful and a bit cheaper, but honestly for a few bucks more for a 5.3 you get an engine with that much more potential. You really just as well get a 5.3 instead. 6.0s and 6.2s are more spendy and harder to find but cubes are king. The biggest downfall of a 4.8 is it has to rev higher to make a little less hp than a 5.3. Keep in mind, if you drop an 8000 rpm 4.8 in a Chevette or Vega with 4.56 gears you'll have a nasty street machine. Put that motor in a 3500 lb wagon and not so much. Also a high revver will wear out or break sooner than a larger higher torque lower rpm motor. I run 406 SBCs in all my cars because I can make 500 ft lbs and shift at 5000 rpms all day long. A motor like that will last 100,000 miles without a glitch, and it'll smoke any small motor on the street because playing with friends often doesn't (and shouldn't) go much beyond first gear.
You can get a decent price on a crate 350 long block if you shop around, and get decent hp. Sometimes a solid inexpensive crate motor and a cam swap will put 300+ hp to the wheels. That's not too bad.
When I first started building cars at 16 I had a 1/4 mile time in mind and then built accordingly. My car was a 73 4 door Nova that ran 14s in high school with a fairly mild 327 (similar to Ebrock Performer parts). Then I saved some money and a few years after high school built a 406 motor and suspension that pushed the car to mid 11s at the track, a computer dyno program said it should be roughly 560 hp. I drove that car on the street everyday and it was a blast.
As per your dad's worries, I think a 13 second car is reasonable and respectable for a teenager. Keep in mind if you have stock 180 hp or 700 hp, either way you can get hurt or into trouble if you don't respect that your piloting a 3500 lb torpedo. As I said I drove my 11 sec Nova that I built for street racing (like you see on street outlaws) and only got out of control once when I hit a puddle from a lawn sprinkler and I happen to have had drag slicks on the car at the time. The way you drive is key to safety rather than how much hp you have. A few key rules will keep you safe. Don't be stupid. Don't drive in a normal crazy teenager way in the rain or snow with power under the hood. Take it easy so you don't blow the tires away and slide the rear out. When it does rain or snow, find a large parking lot and purposely try to get the car out of control and practice getting it back under control. If it has enough power to spin tires on dry pavement practice launching in parking lots and learn how to control wheel spin. Practice also launching in a way that doesn't spin, you'll find you'll smoke your friends off the line every time because they only know how to put the pedal on the floor and lose traction. Don't drive stupid in traffic or neighborhoods! There is a time and place for everything. If you can start autocrossing, I think that's the safest and most fun you'll ever have becoming a better driver. Every teenager should autocross, you become more responsive and alert, you think fast and react to quick changes in driving direction, and you become incredibly familiar with your car and how far you can push it and still maintain traction and control. Recently my son was driving my wife to the store and a car pulled out where he was driving. My wife told me later she was so impressed, he didn't even think about it, he simply changed lanes and went around the other car and kept driving much like dodging cones. He avoided an ugly accident because autocrossing has made his "defensive" driving near habit and unconscious.
Good luck with your decisions. If you're unsure of a decent street car set up please ask here, us old guys have been there and done that and are happy to help you.