I have a basic 18-55mm kit lens. I am fully happy with this.
If this is the case, then why "upgrade" at all? Why not use your 18-55 until it frustrates you. The frustrations will point to what you might want in your next lens, so long as you're sure your technique is good. A lot of folks like to blame the 18-55 when it's really bad technique that's at fault. (See: Why are my photos not crisp?)
Now I am interested in upgrading my lens to have higher image quality.
You must be more specific about what you mean by "higher image quality." If you mean less chromatic aberration, better sharpness, better contrast, less distortion, then maybe a lens upgrade can help you. If you just mean "prettier pictures" then you may need to have a solid rethink and consider whether this is really a gear issue, or a technique/knowledge issue. Owning a dSLR doesn't make you a photographer. It just makes you a dSLR owner.
I noticed that the Canon 24-105mm L lens can be used for multiple purposes like portrait, landscape, and some low level of macro.
Actually, not really any more than your 18-55 lens, and in the case of landscapes, possibly less. Both the 18-55 and 24-105 lenses are good as walkaround lenses, but are both possibly too slow for portrait work (maximum aperture of f/2.8 or wider is generally preferred for portraits), too short for sports or wildlife shooting, and as zooms may have more image quality compromises to accommodate the zoom range.
The 24-105L, actually, among full-frame folks, is seen as a "compromise" lens--it is the Canon full-frame kit lens. Its zoom factor is ~4x, and most folks prefer a factor of 3x or smaller for image quality performance and fewer compromises. It also has a maximum aperture of f/4 which is simply middling--not fast, not slow.
Also I have seen that this one is wider than my kit lens.
It's not wider. The smaller the focal length, the wider the lens. You have an 18mm lens. The 24-105 is, at its widest, 24mm--which is narrower. This is the main reason it's not recommended for a crop-body shooter. The 24-105 is wider on a full frame body than the 18-55 is on your 70D. But to get an equivalently wide field of view, on a 1.6x crop APS-C sensor, you need a 15mm lens, like the EF-S 15-85mm IS USM, which is often touted as the crop analog to the 24-105L.
dSLR lenses work better as special purpose tools than general-purpose ones. If you want a portrait lens, get a portrait lens (e.g., 50mm f/1.8 STM or 85mm f/1.8 USM). If you want a landscape lens, get a landscape lens (e.g., EF-S 10-18 STM). If you want a telephoto lens, get a telephoto lens (e.g., EF-S 55-250 IS, or EF 70-300 IS USM).
Is it good to move to this L lens or can I purchase a Canon 70-300mm IS STM lens which is of little lower price or Canon 70-200mm L lens (without IS which is bad for me). Which is a better upgrade for image quality and multipurpose use for this price range?
Nobody can tell you if this is a good move or "worth it" for you. Only you know what your budget is, and what and how you want to shoot. The answer you get will depend heavily on those factors. A walkaround zoom and a telephoto zoom are not interchangeable lenses that can do the job of the other. How you frame, what type of working distance you like, and what you're shooting will determine if a walkaround zoom or a telephoto zoom are "best" for you, but most of us would say they don't overlap, and if you need both, you should get both.
dSLR lenses are expensive. That's just the way it is. Attempting to save money by having "multipurpose" lenses is doable, but will inevitably involve some form of compromise, either in image quality or usability. There's kind of no way around this.
See also: Lens upgrade paths (sub $1000) for the EF-S 18-55mm IS kit lens for Canon APS-C cameras