Why do my solder joints look like little blobs?

You've basicly answered your own question. The joint should look like a small volcano or cone wrapping the lead, and you should still be able to see the outline of the pad and lead under the solder. If it looks like a fat ball sitting on top of everything, you've used too much. The later joints you described are what you want. A trick that helped me a lot was feeding the solder into the junction between the tip and the pad, not onto the tip itself. If the solder melts the instant it touches the iron, you're heating the solder instead of the joint, and you end up with a cold-looking lump even though it seems fine. Keep the iron there until the solder flows into the pad and around the lead on its own, which usually only takes a second or two once the joint is actually hot.

On the slow-melting iron, that's normal for a basic stick iron rather than a sign it's dying. Cheap unregulated irons lose heat fast when you touch them to a pad, and they take a while to recover. The bigger cause of sluggish melting is the tip. Old burnt tips transfer heat terribly. Clean it on a damp sponge or brass wire cleaner, then melt a little fresh solder onto the tip so it's shiny before you touch it to the joint. A well tinned tip makes a huge difference. Same with the joint itself: a bit of flux makes solder flow like it's supposed to. Even a cheap pen of flux paste will cut down how long you have to hold the iron on there. For the blobby ones already on the board, you can fix them. Add some flux, hold solder wick over the joint, and put the iron on top of the braid. It'll pull the excess off. Then flux again and reheat the joint briefly so it reflows into a proper shape. Much easier than trying to start over. One more habit worth building early: solder for two or three seconds, stop, and let the joint cool without moving the lead. If you wiggle anything before it sets you get a dull grainy joint.

Agreed on the tip cleaning part, that made the biggest difference for me when I started out. I used to think my iron was weak too, but it turned out the tip was blackened and the solder just sat there. I started tinning it before every session and joints that used to take ten seconds of contact now flow in a second or two. Also cleaning the legs of components with isopropyl before soldering helped the solder wet the pad properly instead of beading up.

What finally clicked for me was switching from a wet sponge to one of those brass coil tip cleaners. With a sponge I was always cooling the tip down right before each joint, and my little iron coud never recover in time. With the coil I just stab the tip in for a second and it's clean and still hot. Joints got way faster and the blobs mostly stopped once the tip wasn't half cold every time. Also, don't snip the leads flush right away, the joint is still settling and you can crack it if you cut too soon.