As of now, I have my pictures (shot on a regular lens) which I stitched together using Hugin. What I got is a long panorama (similar to something that can be shot using the panorama mode on a smartphone) but I don't really know what to do with it. I used Panotour in order to create the tour and put it on a website but I don't get that immersive/fish eye kind of feel, basically it's just like I was scrolling through that panorama. What am I missing? Thanks so much!
What you're missing is a lot of scene coverage. To get a 360ºx180º full spherical view, you have to cover the entire sphere. Without a fisheye lens, this is likely to mean multiple rows as well as a zenith and nadir shot, with precise coverage.
In hardware terms, this probably means you need a panohead on a tripod--or at least a plumbline--to make sure you rotate somewhere close to your lens/body's no-parallax point in space, both in pitch and a yaw.
Fisheyes are most typically used because they can easily yield more than 130º of HFoV (horizontal field of view--coverage across the long edge of the frame). A rectilinear lens has a lot smaller coverage. So, for example, if you're using an 18-55 kit lens, zoomed out to 18mm, the coverage you can get on an APS-C body is typically HFoV: 64º; VFoV of 45.2º (see Bob Atkins's FoV calculator).
So, if you hold the camera in portrait orientation, to cover the 360º in yaw, you'd need: 360/45.2 => 8 images around in yaw, and to cover 180º in pitch (from straight up to straight down), you'd need 180º/64 => 3 rows for each of those 8 images. And that's without any overlap, which you require for stitching. If you assume 25% overlap, then that's more like 3 rows of 10 images each plus a zenith (straight up) and nadir (straight down) shot.
So you can see why having a panohead with marked detent stops every 30º or 45º or so to track coverage, or a much wider lens might be a good thing for this type of shooting. Conversely, shooting with a rectilinear lens does get you a much higher-resolution final panorama, so there is a plus side to having to do all the additional stitching.
What you are missing is the type of projection used to create a 360 degree tour. The software you are using seems to be building a cylindrical projection and then unrolling it. In other words, it is blending the images together to remove seams along the sides, but it maps them on to a flat circular wall around the viewer.
This results in a very natural, and I'd argue preferable, perspective that looks naturally like you are turning around when you rotate through it. It does not, however, do a good job of capturing the top or bottom of the space as it is a fixed perspective looking out at the horizon.
If you instead use a spherical mapping, you will get the look you are talking about, but will additionally need photos that cover the top and bottom of the space so that you completely cover the entire sphere that is being mapped to. The "fish eye" effect you get when viewing these is because of the mapping of the sphere back to a rectangular viewpoint. If you view one of these spherical mappings from a 3d viewer (such as a VR headset), you'll instead see it as a 3d sphere that you are looking at and it won't seem like it is fisheyed, but rather feel like you are looking at the cylindrical projection on a flat 2d screen.