normal = {1,2,3}/Sqrt[14];
r0 = {1,2,3}/Sqrt[14];
ContourPlot3D[{x^2 + y^2 + z^2 == 1, ({x,y,z}-r0).normal == 0}, {x, -1,1}, {y, -1,1}, {z, -1,1}, ContourStyle -> Opacity[.8], AxesLabel ->{x,y,z}]

Here I consider the surface x^2 + y^2 - z^2 + z^4 = c.
 I start with a point, r0, and the level surface then plot the surface through that point, and the tangent plane at that point.
Variety and Manifold

r0 = {.3,0,.5};
normal = {.6,0,-1+.5};
ContourPlot3D[{x^2 + y^2 - z^2 + z^4 == .3^2 -.5^2+.5^4, ({x,y,z}-r0).normal == 0}, {x, -1,1}, {y, -1,1}, {z, -1,1}, ContourStyle -> Opacity[.8], AxesLabel ->{x,y,z}]

We can plot two surfaces and see their intersection.

ContourPlot3D[{(x-.5)^2 + y^2 + z^2 == 1, (x+.5)^2 + y^2 + z^2 == 1}, {x, -1.5,1.5}, {y, -1,1}, {z, -1,1}, ContourStyle -> Opacity[.5], BoxRatios -> Automatic (* try commenting out the BoxRatios *), AxesLabel ->{x,y, z}]

ParametricPlot3D can plot curves (1 parameter) or surfaces (2 parameters) in 3-D space.

ParametricPlot3D[{Cos[t], Sin[t], t/Pi}, {t, 0, 4Pi}, AxesLabel ->{x,y,z}]

ParametricPlot3D[{r Cos[t], r Sin[t], t/Pi}, {t, 0, 4Pi}, {r, -1, 1}, AxesLabel ->{x,y,z}]

You can use Greek letters!  And Pi or the Constant π.  The “Assistant” palettes allow this

ParametricPlot3D[{Sin[ϕ]Cos[θ], Sin[ϕ]Sin[θ], Cos[ϕ]}, {ϕ, 0,π}, {θ, 0, 2π}]

Plot[ {-x, Log[x]}, {x, 0.1,2}]

You will need to do something very much like this on homework problem 6 of HW2.

FindRoot[-x == Log[x], {x,2}]

soln =FindRoot[-x == Log[x], {x,2}]

x/.soln

Plot[Sin[x t]/.soln, {t, 0, 10}]