Grower
Growers are what keep your fortress alive. Everyone needs food to live, and the main way to make sure there is always food is to grow crops. Because of this, growth is one of the most important skills to learn when starting over. A skilled grower can gather and plant much more quickly and in larger quantities. This speeds up all food-related businesses, such as brewing and cooking, like a butterfly effect.
One skilled grower can beat multiple unskilled dwarves. Since farming is a reliable way to get food that won't run out, you should definitely take growing as one of your first skills.
Miner
Mining is almost as important as being able to grow. Dwarves live in the mountains, and the only way to build your castle is with miners. Any mining chores that you give to specific miners will be done right away. On a new run, their job is very important because your dwarves won't have a place to live until they can mine one out.
Most likely, you'll want two of the seven dwarves you start with to learn the digging skill. They will have to do a lot of work to make the cold stone into a nice place to live.
Mason
Without furniture, a castle isn't complete, and that's where masons come in. Dwarves with the construction skill are asked to make things out of stone, like furniture and buildings. Masons will never run out of rocks to work with because they will have so much to choose from.
Masons make things like trade depots, bridges, furnaces, and wells that are very important. Stone blocks are also used to build walls and other defensive buildings. Masons are important builders in a fortress because there is so much equipment to take care of.
Cook
Dwarves are tough by nature, but they can only eat raw food for so long before they get sick of it. Luckily, if you bring a cook with you from the start, your people will be happy, healthy, and never go hungry. Making meals out of raw items will increase the number of food sources that can be eaten. Raw food doesn't last as long as meals that have already been made.
Cooks can use any animal fat to make tallow, which is a key ingredient in making soap. Giving your gnomes a good meal and a bath will do a lot to keep their spirits up.
Mechanic
Most of the time, the world of Dwarf Fortress is hostile, so you should get your fortress ready for possible strikes. With the mechanics skill, you can make traps that will stop any invader. Mechanics can make things like pressure plates, levers, and shutters, which are all very complicated. If you set up enough traps, any nasty goblins who try to attack you will feel like they are in a Saw movie.
Even though being a mechanic might seem like a niche job, you won't be sorry if you learn the skills before you start.
Carpenter
Beds: It's easy to forget how important these wooden structures are to society. People spend half their lives sleeping, but no self-respecting dwarf wants to sleep on the floor. In Dwarf Fortress, beds are just as important as they are in real life. The catch is that they can only be made out of wood in Dwarf Fortress.
So, the building skill is a bridge to a good night's sleep. Carpenters make things out of wood, like bins, jars, and, most importantly, beds. The first thing you should do when you start a new game is chop down trees and make beds.
Wood Cutter
One of the first things you can gather is wood. It is also a very important part of your fortress. Because wood is so important, it makes sense to have a dwarf start with points in the wood cutting skill. Every point they spend makes it faster for them to cut down a tree. This saves a lot of time over time and gives your dwarf more time to do other things. Wood is important for things like machine parts, ash, and fire, as well as beds.
Charcoal is a fuel source that every new castle needs, so the woodcutting skill is very important. Ash is also a useful material because it is hard to get and is used to make soap and fertiliser.