Basic method for 2x oversampling?
Interpolation (sampling frequency 44.1 kHz ➔ 88.2 kHz)

Your original 44.1 kHz sampled signal has frequencies up to 22.05 kHz, so you should lowpass filter at 22.05 kHz after dilution with zeros. Your filter should have a gain of 2. Otherwise the signal amplitude drops to half because you set half of the samples to zero. Like Jim Clay says, you can combine these things; there will be plenty of opportunity for optimization.

Downsampling (sampling frequency 88.2 kHz ➔ 44.1 kHz)

If your 88.2 kHz sampled signal is the one you lowpass filtered at 22.05 kHz, you can simply decimate by throwing away every second sample. Otherwise, first lowpass filter at 22.05 kHz to prevent aliasing, and then decimate. If you combine lowpass filtering and decimation, you can optimize again.