Your instructor is primarily responsible for making sure that you meet all FAA requirements by the end of your training, so you shouldn't need to worry that much about the details. However, if you're curious about what your training might look like, here is a basic overview:
Part 141 Flight Schools vs Independent Instructors
In a broad sense, flight training can occur under one of two different sets of rules, Part 61 and Part 141. Training with an independent instructor is usually the more flexible of the two, and will occur under Part 61 rules. You can think of it like hiring a private tutor. You meet with your instructor to develop the required skills and knowledge at your own pace and on your schedule. You don't need to worry about doing your training only at certain approved airports with specific and often arbitrary school policies (like uniform requirements). Part 141 training, on the other hand, is more like going to an academy with a set curriculum, lessons done in a particular order, and with periodic internal tests (called stage checks) throughout your training that you must pass before you can advance. You'll need to work with your assigned instructor within the school's scheduling and training policies, even if they seem arbitrary or unnecessary for your particular aviation goals.
Probably the main advantage to Part 141 training is that, in theory, you should receive at least a minimum quality of training due to the structure and approval process the school and its curriculum must go through. Horror stories you might hear about someone's crazy flight instructor making them do 40 laps in the pattern at 3am to "build endurance" aren't going to happen in the Part 141 environment. There are technically also some minor reductions in the amount of required hours you need, but practically speaking, almost everyone goes beyond those hourly minimums anyway. The downside is, you will often only get the minimum quality training because a flight school is ultimately a business with an incentive to have as many paying students as possible, and if it takes those students longer than necessary to complete their training, that's just more money for the flight school. Although there certainly are high-quality, reputable schools out there, for some, failing stage checks and redoing lessons due to bare minimum instruction is commonplace and just means more revenue for the school. Additionally, the required structure of the approved courses can lead to inefficiency for individual students. Are you a fast learner that wants to jump ahead to later stage training before you've passed your first stage check? In cases like this, you'll be required to wait, maybe weeks, to pass your stage check before you are allowed to continue under the Part 141 structure.
The biggest advantage of doing your training with an independent instructor under Part 61 is the flexibility it offers, but your experience is also greatly tied to the quality of your instructor. A good independent instructor can get you through your training efficiently, often for less cost than a flight school program, and make the whole process feel more personal and enjoyable for you. After all, flying should be fun! But, if your drill sergeant instructor thinks you need to do 40 laps in the pattern at 3am to "build character," there aren't any school policies stopping them. From this perspective, you are a client, and your enjoyment of the training experience matters, but your instructor is also the one personally responsible (ethically and legally) for making sure you are a safe and effective pilot in the end, so you will need to demonstrate knowledge and proficiency to a level your instructor deems sufficient in order to earn their endorsement. Finding the right Part 61 instructor should result in a good balance between your needs as a learner and their professional needs as a safe and effective CFI.
The private pilot certificate is usually the first one you'll get (although there are Sport and Recreational Pilot certificates, that is another topic). It allows you to act as the pilot in command of an airplane day or night, by yourself or with passengers, at your discretion. You can take your family on a flight to visit relatives, take your date out on a Saturday evening to watch the sunset, or go fly around by yourself on a Tuesday afternoon just for the fun of it. The main things you cannot do with just a private pilot certificate are (1) receive compensation for your flights (with some very limited exceptions), and (2) fly in or too close to clouds or bad weather. If you are happy being financially responsible for the cost of your flying and only want to fly on nice weather days anyway, then a private pilot certificate may be all you really need.
Basic outline of training required:
At least 40 hours of flight time
At least 20 hours of training with an instructor, and this must include:
3 hours of cross country flight
3 hours of night flight, and this must include:
A cross country flight over 100 miles
10 full stop takeoffs and landings
3 hours of basic instrument maneuvering
3 hours of practical test preparation
At least 10 hours of solo flight, and this must include:
5 hours of solo cross country
A cross country at least 150 miles with full stop landings three different points
3 full stop takeoffs and landings at a control towered airport
An instrument rating allows a pilot (private or commercial) to fly into the clouds or weather conditions worse than those required for basic visual flight. It isn't really a certificate on its own, but gets added to your existing pilot certificate as a rating, and is usually the second course of training a student would pursue. Even if you only intend to fly on nice sunny days, an instrument rating can be valuable. It also allows you to flight in Class A airspace above 18,000 feet (which might be applicable to you depending on the capabilities of the airplane you are flying), and it enables you to operate on IFR flight plans (which can be useful in many circumstances, even if the weather is nice). The additional training and challenge of controlling the airplane only with the instruments also tends to make you a better pilot, even in visual conditions, and for that reason, I always recommend that students consider getting an instrument rating, regardless of what kind of flying they plan to do.
Basic outline of training required:
At least 50 hours of cross-country time as pilot in command
At least 40 hours of actual or simulated instrument time
At least 15 hours of training with an instrument instructor, and this must include:
A cross country flight of 250 miles with an instrument approach at each of 3 different airports
3 hours of practical test preparation
The commercial pilot certificate is required for you to receive compensation for your flying (still with certain limitations). Much of the content of commercial pilot training is similar to private pilot training but with tighter tolerances, more depth, and focus on precision. It may be the final certificate earned, unless you are planning to fly for the airlines or conduct cargo operations in large transport category aircraft, which typically require an ATP certification. Depending upon how much time a student needed to get through their private training and instrument rating, a certain amount of time building may be necessary to meet the minimum total time requirement. It is also worth keeping in mind that holding a commercial pilot certificate does not automatically authorize you to operate as your own little miniature airline or charter company, flying customers around in your airplane and charging them airfare. The rules governing commercial pilot privileges and limitations are complex, and learning specifically what you can and can't do as a commercial pilot is a major cornerstone of the training.
Basic outline of training required:
At least 250 hours of flight time
At least 100 hours of pilot in command time
At least 20 hours of training, and this must include:
10 hours of instrument training (but instrument rating training can be credited toward this)
10 hours of training in a complex airplane, turbine airplane, or technically advanced airplane
One 2 hour daytime cross country of at least 100 miles
One 2 hour nighttime cross country of at least 100 miles
3 hours of practical test preparation
At least 10 hours of solo time, or time performing the duties of pilot in command with an instructor, and this must include:
A cross country flight of 300 miles or more, with landings at 3 different points
5 hours of night flying with 10 takeoffs and landings at a control towered airport