The Illustrator ellipse tool can be found in the toolbar beneath the rectangle but you can always create a custom set and access there if you wish.
The new 'live shape' status was added to the application a while ago so you probably should have that feature but if you have an old version then the feature may not be available. The live shape has some new features added such as the creation of arcs etc. Select it. Drag and create on the art board. Set the fill and stroke etc and window and properties panel. You can modify the position as well as the weight and height as well as flip and change the angle (though no different if it is a circle) as well as the pie slice settings such as start and end
The first thing to notice after selecting it and creating the artwork is that as you hit a circle shape as two cross hairs will appear and disappear as the size of the artwork changes - they will appear if the width and height are equal. If you hold down the shift key, you can create a circle but the smart guides will not appear. Once you have the shape, you can then manipulate it either via the bounding box etc or via the transform panel.
There are a two sets of fields for the height and width as well as the ellipse height and width, they are not equal but they can be set to the same but only when the shear angle or angle as it shows the weight and height as well as the weight and height of the bounding box which is not the same thing when angles are involved. You can change the width etc via the up and down controls on the keyboard as well as just typing in the value or using the bounding box or the side points (to size proportionally use the left and right and use the top and bottom ones to squeeze or stretch up and down. You can change the angle (but if this is a circle then you won't see any difference).
You can use a combo of the properties panel and the layers to create a quick contour design. Select a circle say 600 x 600 and then go to the layers panel and select that and go to the right side menu and select the duplicate command and then go to the transform and height and width field and enter 500 x 500 and go to the properties panel and change the color. Repeat the duplicate and transform and re-color with 400 x 400 etc. You can also enter a percentage into the transform such as 99% of the height and width as well as adding in a mathematical equation such as the current value - 2 etc
This is one of the weird features, if you change shear value then the vector artwork will be sheared and you can see it stretch off at 45 degrees etc but as soon as you enter a value, the transform panel sets the shear back to 0. In earlier versions, shear would break the live shape but that does not appear to be the case now.
You may or may not love the live feature. You can expand the artwork by going to the object menu and shape and expand and you can then also re-convert it back again via the same menu. Once you have the normal path then you can use the sliced path (if the pie slice was used) in numerous ways such as creating all kinds of abstract backgrounds. You can also create all kinds of other artwork using effects such as the transform with the path. To do the reverse, select an expanded design and go to the shape and 'convert to shape' command. You will note once the artwork has been turned into a regular circle then some of the controls in the properties are gone. When you run the convert to shape then they will return
You can add all kinds of transformations and other effects to the artwork. Select and create a slice using the slice angle 0 to 5 say. Go to the effects menu and transform commands and set the preview to ON (it is not sticky) and set the number of copies to 20 or 30 and then set the angle and rotation center as well as the move setting
Select the design and set the fill and stroke and then go to the appearance panel. Select the path entry and then go to the right side menu and select the new stroke option and go to the bottom entry for the stroke and increase the size.
You can use that to create a very quick burst / zoom effect. The transform effect will appear in the appearance panel. You can then change the setting by double clicking on the transform line. Use a small path (still editable) along with a distort / transform added with a number of copies and combined with a drop shadow. The entire artwork is still live and can be changed at any point. You can also add additional transforms such as only setting the number of copies and the scaling to create interesting radial artworks. The artwork is still editable so you can create thin or thicker slices interactively as well as via the transform panel. You can also use other effects as well such as blurs and distortions and zigzags. Select it and go to the window menu and properties and go to the properties panel and create a circle using red as the fill and set the pie start to 0 and pie end to 10. Effect menu and distort and transform and the transform command and preview ON and set copies 10 and set angle to 20 and set the center of transform to top left and click OK. As it is a live shape then you can alter the angle etc and the change will be reflected in the transform as well
You can set the shape as an arc by using the pie settings with a shape that has no fill and has a stroke. You can use the direct selection and delete the 'middle' point of the shape by pressing the delete key
You can create a path and distort the design etc and then with it selected, go to object menu and repeat and radial (or grid etc) and you will then see the design repeated around a central point. You can add more instances as well as change the size of the circle and also double click the object and edit it such as change the size as well as re-color or duplicate the artwork etc and create all kinds of unique designs. Of course, you can also fill the artwork such as a gradient as well as add a stroke
If you have Astute graphic plugins then you will have the AG splatter plugin and that can be found in the window menu. Select an object. Click the apply AG Splatter and then click the seed. Set the density as well as the throw, spread etc of the artwork. You can set the color to use the existing color of the path but you can specify a color and opacity. You can modify the shape as well by setting the value to a low value to create blots. You can use this on multiple selected shapes. You can create all kinds of dramatic designs around the edge of the vector. You can also combine them together and then apply the plugin for even more hi-impact designs.
Draw and hold down the shift to create a circle, set the fill to yellow and set the stroke to black etc and go to the effect and 3D and inflate for an instant spherical shape. Go to the materials tab and drag over one of the materials such as sand dunes or leather etc and now you have a leather sphere. Go to the lighting tab and set the lighting as required. The design is still alive so you can always extend the hight etc to create an egg shape. You can't rotate all the way around because it is only one side of a sphere which is hollow if rotated around to the other side unless you click the check box for the inflate both sides. If you want to remove this at any time, go the appearance panel and right side menu and clear appearance.
You can create two shapes, the first big and the second slightly smaller. Select both of the shapes and use the alignment tools in the control bar to align to the center / middle and you can then go to the pathfinder panel in window menu and pathfinder and select the minus front to create a ring design. You can now fill that with gradients, add 3D effects etc