More specifically, how extensive is my knowledge of how steel and bronze work. Do I understand that iron has gaps that are filled by carbon in the production of steel? Or is it I just know it works?
The first sentence of the top answer makes the point that blacksmiths of that era were primarily tool makers, not armourers.
You probably wouldn't be making arms or armor. Arms manufacturing was a highly-skilled and highly professional trade that generally took place in specialized private workshops, court workshops, or military/garrison armories.
The average blacksmith would have been making more mundane metal goods. In rural areas, you'd be making horseshoes (and you might be expected to act as a farrier (i.e. the person who shoed horses), metal tools and fixtures. You'd also fix farm equipment and farming tools. In urban areas, you'd be making/mending a similar variety of metal goods (although obviously not farm tools).
Keep in mind that during the renaissance period, there was no concept of modern elements. Those that studied such things tended to subscribe to the ideas of Aristotle, that there are only four or five elements (earth, air, fire, water, ether/quintessence). But you might not even know that much. You don't get much training in science. Science is barely even a word in the Renaissance. They would call it "philosophy". You didn't go to school, and there is a chance that you could not read or write, though it is more likely that you could than a farmer.
There is very little in terms of period writing to show that anyone, blacksmith or no, even clearly understood the difference between iron and steel. They were aware that some metal could be quench hardened, and some metal could not. But if there was a known process for converting iron into steel in Europe, it was kept as a closely guarded trade secret, and likely not written down.
By the Renaissance, blacksmiths would be obtaining metal through a combination of early iron mills and by reusing scrap.
A blacksmith would know of bronze, but wouldn't likely work with it himself. Bronze was no longer being used for blade edges at this point in time, so there was no point in work-hardening bronze objects. Bronze was pretty much left to being used for making cast objects. And this would be done by foundrymen, not blacksmiths. The ability to cast bronze was still important, particularly because Europe did not work out the ability to work with cast iron until later in the Renaissance (China had it centuries before Europe).
This is a little confusing to most people, but "cast iron" in fact has more carbon in it than high-carbon steel. This carbon is critical for iron to be castable. Without it, iron brought up to melting temperatures immediately "burns" or oxidizes when oxygen is present. The result is brittle worthless iron-oxide. But I go off topic.
You know that there is a relationship between the color that iron glows and the temperature it is. (it starts at a dark red, to bright red, then orange, yellow, and finally white). You know that above white, the iron sparks and burns.
You know that there is a grain structure to wrought iron, and you know how to take advantage of that in order to make the strongest piece possible (many modern blacksmiths don't know how to do this). We know that fact is true by studying the grain patterns in renaissance iron artifacts. For example, the smiths would often stretch out and fold over a bar, welding the ends together to make a hole. This takes much more work than just punching a hole in a bar and drifting it to the needed size, but in wrought iron, doing that would create a weakness.
Again, depending on where you are in the renaissance, you may know that some iron can be quench hardened while other iron cannot. If you know this, then you also know how to most effectively fasten that steel on to the cutting edge of an iron tool. You could theoretically make the entire tool out of steel; but it was so much more expensive that doing such a thing would seem quite silly to you.