Re: How can I increase (or decrease) the frequency of alternating current (ac)?

Probably the simplest way is to use a bridge/full wave rectifier.  That 
will double the frequency.  If you are looking for a way to create various 
frequencies from a fixed frequency AC supply, that's quite a bit more 
complicated.

You are correct in that by the proper choice of simple circuit elements an 
LC, "tank circuit" can produce varying frequencies, but for control 
purposes, the LC circuit is usually driven from a DC supply.  Connecting 
it to an AC supply would create wild interactions and harmonic "beat" 
frequencies, or if the resonant frequency of the LC pair is incompatible 
enough, no oscillations at all.

Also, in dealing with wall outlet voltages, the power handling and 
insulation values of the components in any circuit becomes a 
consideration.  For "high" power applications, frequency control is not 
attempted using solid state components.  A constant speed motor is used to 
drive a generator through a variable speed fluid coupling.  Varying the 
rotating speed of the generator produces the varying frequency.  That type 
of frequency control has been employed for controlling flow through the 
cooling water pumps at the Tarapur nuclear plant.

The Tesla coil is a transformer, and like all AC transformers, transforms 
the voltage without affecting frequency.  Outside the lab, Tesla coils and 
Jacob's Ladders have little practical application - the most memorable of 
which is their use in Frankenstein movies.

So assuming that you really want to do this using wall outlet power, 
first, using a transformer, reduce the voltage to something easily handled 
by readily available (cheap) components.  Second, covert it to DC.  Then 
use the DC to power a tank circuit containing LC components that resonate 
at the frequency required, or even with variable L and/or C components to 
allow adjustment of the frequency.

If you want to get into more complicated solutions, crystal oscillators 
that generate the Megahertz in a computer, Zener diode circuits, and the 
kind of circuits dreamed up be graduate Electrical Engineering students 
can lead to unique, exotic, and probably impractical answers, but they all 
seem to start with a DC power supply.  And, all of these are low power 
circuits.  For power applications, the old Motor-Generator set is still 
the cheapest and most reliable.