Rule Number one on this forum Is never ask a question that you want a straight forward answer on and never ever say you are new! People on here will happily spend time to type out all the ways you are horrible for ever asking but never spend the same time and effort to answer you.After all why ask a simple question and hope that someone will take 5 minutes to answer it to help save you countless hours and hours of research. Why would anyone on a machining forum want to answers the most basic of machining question? It's exactly like an elite shop that only hires journeyman. They are happy to pull from the pool of journeyman but never contribute to it by teaching an apprentice. That should be done in the someone else's shop. Only come to this forum once you have all the answers and no questions then you to can rip into anyone who asks a basic question. That is your final step to joining this elite club
Now lets see if my limited knowledge. I don't know what your budget is so you can adjust this to suit you. Lets start with the lathe. Since you don't have a tool post that's the first step and really for starting out is based on budget.
A lantern tool post is probably the very cheapest but is a pain in the but to use especially as a beginner I have a few and personally would avoid that option.
Next up would be a 4way tool post this may also have other names. It's basically a cube of steel with a slot on all 4 sides that you tool goes into and is clamped with set screws from the top. it is held onto the compound with a large bolt/stud through the center into a tee nut. There are pros and cons to this set up as well. The biggest pro is that if your on a budget it is a great beginner project and depending on what you have available there are many simple ways to build it. Two ways include either starting with a solid chunk and milling out a slot big enough for you tools plus some extra space for shims to get the tool on center. Or make it out of 3 thick pieces stacked up the center piece being a smaller square than the top and bottom and bolt it together. This type of tool post needs to be sized to your lath so the slots put your tool on center usually with a few shims. It is easier to use than a lantern tool post but not as easy or convenient as a quick change tool post.
And this brings us to the quick change tool post. This is where budget come in. The sky is the limit to how much you can spend starting at decently priced and going up to the moon form there. There are many types and styles to choose from and pros and cons to each but it really comes down to personal preference. The Aloris style is the only one I am familiar with. You can spend the money on a actual Aloris or there are other cheaper options some of them are cheap and some are perfectly adequate for the home shop. They come in to types wedge type and piston type. the piston type is cheaper but not as rigid I believe. The wedge type is what I have and is usually the preferred type if budget permits. The quick change is nice because every tool is easily adjustable in height and once it is set it is repeatable every time you remove and install or switch between tools.
Now onto lathe tools.This I will break down into 3 type High speed steel (HSS), brazed carbide, and carbide insert tooling.
HSS has been used for decades and decades ever since its discovery in that year it was discovered
This is the one I recommend. It requires a bit of learning to grind the tools to get the results you want but that is very valuable and useful knowledge to learn. It is the cheapest option and offers an infinite amount of possibilities. I started with HSS when I got my first lathe 12 years ago and it is still my go to for lathe tools.
Next up is brazed carbide. basically I look at it as a cross between HSS and carbide insert tooling. It is a piece of carbide brazed to the end of a square steel shank usually the same size as the common HSS blanks. These can be sharpened on a special grinding wheel. They come in preformed shapes ready to cut. cheap ones sometime need a bit of touch up on the grinder with the special wheel. I have some I use sometimes for chamfering.
And then there is carbide insert tooling. This uses carbide bits that are bolter of clamped into the end of a tool holder. they come in a variety of shapes and sizes and types. every shape of insert requires its own holder. If you are going this route my advice is research they best types to suit your needs and then buy used until you find what works for you and what you like. I don't know enough about this type of tooling to make a suggestion. But others on this site do and can hopefully steer you in the right direction.
On to the mill.
End mills come in many types and sizes all with there uses. Or so I am told as I have not yet tried every size and type. Some say for beginners and hobbyist that cheap import end mills work. This however has not been my personal experience. Your mileage may vary. I recommend buying a few decent quality ones in the sizes that will best suit you needs and start with those. Or picking up some used good quality ones if you can find some at a good price. They also come in HSS and carbide. I have no experience with carbide end mills yet. But my understanding is they stay sharp longer but are harder and therefore more brittle and more expensive. Hss end mills can be sent out for sharpening or sharpened at home if you feel ambitious and want to build a jig for that. I would imagine that carbide cold be sharpened as well but I don't know for sure. End mills also come in 2 flute and 4 flute but that will depend on what your needs are.
Next we can get into larger mills like face mills for when you have a large surface and don't want to make 22 million passes. To start with a fly cutter is the cheapest and easiest route and leaves a very nice finish. They are cheaper than face mills to buy or make an excellent beginner project. They also require less power for their size compared to face mills I believe. Face mills use carbide inserts (not the same inserts as the lathe tooling). Face mills can take a deeper depth of cut than a fly cutter or so I believe. I don't have a face mill because fly cutters have always met my needs.
I am not an expert and not a professional machinist but I was a beginner and asked the same question so I hope this helps. All this information is my opinion and I hope others will way in with their views and correct it where needed. I am a strong believer is asking questions and sharing information. As well as If you pull from the pool of knowledge then at some point give back to it. When I was a welding apprentice it really bothered me the big companies that would pull from the pool and only hire journey men and refuse to train apprentices. Sadly that can be what this forum is like at time. Hobby sites are good but some people want to learn from the best to become better and not from someone who who is at the same level as them so to speak. There are good people on this forum who are willing to share their knowledge you just have to be willing to stick around long enough for them to reply and sift through all the negative comments to find the answers.