I used to live north of the Arctic Circle at the northern edge of town, and frequently saw the northern lights from my apartment window. At the time, I studied and worked on a campus housing the Swedish Institute for Space Physics, and had direct contact with lecturers and researchers specialising in aurora.
You certainly don't need any tour to see the Northern Lights (but you might still enjoy one if you have plenty of money).
It's easier to identify weak aurora on camera for reasons others have mentioned, but it's much more spectacular to see in person because it moves. If you see weak aurora, keep watching. Your eyes will get used to the dark and chances are that it gets much stronger.
Once you're in the right spot, you need about as much knowledge to see the Northern Lights as to see a rainbow. Meaning none at all.
When you're around the Arctic Circle, you don't need heightened solar activity to see the Northern Lights. At normal solar conditions, if you go outside between 23:00 and 02:00, when the sky is clear, if you spend an hour looking at the sky, you have around 70% chance of seeing the aurora. In fact, the auroral arc moves south during high solar activity, so if you're in the far north during a major solar storm, it could even happen that it's visible at 50°N but not at 70°N!
How hard is to for someone who has never seen the northern lights to spot them?
Good aurora is easier to see than spotting a rainbow. You might miss it if you're looking at your smartphone, but if you are looking at the sky, you cannot miss it.
Are they only visible for very brief moments
When there is high activity, it will go on for hours, and it will be highly dynamic. I have seen the aurora dozens of times. It has never happened that I stopped watching because activity stopped. In most cases, I stopped because I wanted to go to sleep or had other obligations, and a couple of times I stopped because it became cloudy.
It will of course stop at dawn or become invisible when clouds move in, but if you start watching at 23:00 it's a long session until dawn, and probably you'll have had enough well before (you'll probably be cold). Although it happened that I went to bed at 00:30 to have others tell me the next morning that at 02:00 it got even more spectacular (of course, the most spectacular aurora happens right after you stop watching, in particular if others continue and will testify for you).
or only visible from certain angles?
The angle does not matter.
If one keeps looking at the sky northwards for several hours, would an aurora be easily noticeable?
Certainly, yes. During dark clear-sky conditions under normal solar activity, around the Arctic Circle, if you watch for an hour, your chances are around 70%. You're already in the north, so the aurora can be anywhere: north, south, or straight overhead.
When looking for the northern lights, does changing positions or directions by walking around or driving affect their visibility?
Assuming clear-sky conditions, no — except that the less light pollution you have, the more spectacular they look. But if the aurora is strong, you can see it even when standing right under a street light.
Numerous articles recommend booking a tour to get a chance at spotting them. Those tours appear quite pricey to me for the few hours that they last.
I guess those articles are paid for by those tour guides. I've seen "wilderness overnight snowmobile tours" that go in circles for hours only to end up to a spot that is less than 100 metre from a road. By all means, book a tour if you enjoy the experience of a tour, but if you just want to see the aurora, don't.
Some advice:
It takes a long time to get dark in the north. An hour after sunset is too early. My best sessions have been while camping and setting my alarm clock to the middle of the night, then crawling out of my tent and staring at the night for a little while. But also when commuting home after work between November and February.
Usually, when watching the aurora, it's cold. During clear-sky conditions, temperatures inland can drop to -15°C already in October. That's very cold to sit still for hours, even with good clothes. September is better, but even a mild night at -5°C might feel cold if you're from a warmer climate. Get a good sleeping bag, find a windstill spot, and lie down outside. While waiting for the aurora, you can enjoy the stars and the stillness of the nature. I doubt any guided tour will take the time to do so, and it will be a memorable experience even if you're unlucky and the northern lights do not turn up that time.
There exist smartphone apps that warn you for aurora activity. There also exist various scientific institutes that operate all-sky cameras. That way, you can set an alarm and wait inside without freezing.
Clear-sky is key. This is the only thing where local guides might add value: they should know the local weather and know, depending on weather conditions, which places are more likely to be clear. They might also have access to measurements that you don't. While living in the north, I was surprised that in October, Northern Norway often had clearer skies than Northern Sweden. I write might, because I don't know if guides actually do that. I think the local guided tours when I lived in Kiruna always went to the same spot, had a barbecue, stayed for a bit, and went back to town (I met them sometimes while cycling around at night). But maybe some guided tours do a better effort.
If you see faint aurora, keep watching. Chances are good that it will get stronger within an hour. Indeed, faint aurora might rather look like thin cirrus, although if you look carefully, you will notice that 1) it is too dynamic for thin cirrus, and 2) you see stars through the "cloud". Clouds can be dynamic, but clouds that you can see stars through on a still night are not usually particularly dynamic. A photo with a long exposure (or very light-sensitive sensor) will clear any doubt between cloud or aurora.
There exist some places with glass-roofed tourist accommodation. I've never booked any, and they probably aren't cheap, but I imagine it should be great for aurora-watching. If you do have some budget, I would seek those out before seeking out a guided tour.