Let them lead. Offer them solids when they start stealing it off your plate. My kids were both interested in solids at 4 months. My friend's kids weren't interested until 12 months. The kids know when their gag reflex has settled down enough to manage solid food. There's a wide range of normal in this area.
For the most part, skip the baby food. It's fine to use it as an occasional convenience, but it's usually overpriced, over-processed, and not very tasty. If you want your kids to grow up liking fresh home cooking, don't feed them overprocessed food from a can constantly when they're learning about food. This tends to lead to picky eating as they get older.
Making baby food is simple. Take something soft you are eating. Mash it with a fork. Add a bit of liquid if required. Give to baby. If they need a dead-smooth texture, they're probably not ready for solids yet. Items that can't be mashed (like beef) may need to be pureed.
Introduce one new food at a time, then watch for reactions for three days. Start with foods that are rarely allergens. Add in potentially allergenic foods later. If lots of close relatives have allergies, take this into account. If mom had any major aversions during pregnancy, consider deferring those too.
Our favorite baby food was avocados. They're easy to prepare, nutritionally dense, and not a common allergen.
The pathogen risk of fresh carrot juice should be exactly the same as the risk of a carrot stick. This might be an interesting way to get around the choking hazard issue and introduce a variety of veggies early. Be careful with excessive amounts of juice, though. Most of their liquid should come from milk or water, not juice. It's pretty easy to create a three-year-old juice addict who won't drink water.
You could also get an absolutely massive carrot (like two inches across) and let the baby just suck and gnaw it. Like a teething biscuit, this would be a choking hazard if the baby managed to break/chew it down to chokable size, so keep a baby with a carrot closely supervised.
Have fun!