Hi Paul,
A via is typically there to flow current from one side to the other, but its secondary purpose is to make solder joints stronger and make better contact. The solder will suck up into the hole and make a very strong connection to the leads inside. They are used on single-sided boards for this purpose.
Your soldering is fine - but with a higher degree of risk (high current, money investment) just be very anal about that solder. I like to clean the tip with a wipe on a wet sponge between every solder joint. Then put a tiny spot of solder on the tip so that heat conducts better off of it, press the tip into both sides of the joint at once to get it hot, then apply solder. I usually apply solder to the opposite side of the pin from the iron - this ensures that i'm getting the heat to melt the solder from the pin and not from the iron. Feed the solder straight in and let the heat spread it out. If it doesn't happily jump where you expect it to, you're short on heat - don't panick, just stop feeding solder for a second and let the heat spread. Don't overdo it though, or you'll burn your board and parts
SM soldering is MUCH trickier - especially if you opted for lead-free paste. Get a good sharp pointy tip for that work, i use a hook-shaped one. Put paste on each of the pads, then use tweezers to place the part and hold it. I like to slowly heat the pads so the paste can get a "starter" hold on a few pins before i start to put pressure and get the paste to flow completely... make practice parts
You could probably find images and cut-aways of what solder joints should look like and what they shouldn't look like.
I'm glad I could skimp on the copper a little! that looks pricy!
Keep it up!