Which swing knots to tie?
If the tree limb is accessible via ladder or climbing, then for the first part (around the tree limb), you want to tie a "round turn and two (or three) half hitches". It's a very common knot because it works. It keeps the loaded line as straight as possible but offers high security as well. Another knot, the "bowline", will work, but if you use it, the loop may stay wide open and on every arc of the tree swing and it will rub against the bark on the limb. Friction is the enemy. You want a knot that will grip the limb tight and not allow the rope to move at all. The two loops plus the half hitches grips the limb tight and only gets tighter.
If the tree limb is not accessible via ladder or climbing because it is too high, then the solution for attaching ropes to the tree limb is a Running Bowline knot, a type of slipknot. A benefit of a slipknot is the rope won't strangle the tree limb as it grows. Note that you can also use this option if the tree limb is accessible via ladder or climbing. Use a weaver hand tossed throw weight and throw line to get your rope fed over the limb. A 12 oz. weight should work fine, but heavier weights are needed for higher limbs. Really, really high limbs may need a launcher if you can't get the throw weight and rope over the tree limb. Once you get the line fed over and back down to you, you will likely need to tie a slip type of knot to adhere you rope to the limb.
Finally, for the second part (through the swing seat), you will want to tie a "double overhand stopper knot"