Real time slope and peak detection and calculation
You should start with a bandlimited differentiator (equivalent to a differentiator followed by a low-pass filter). The differentiator will remove the low frequency trend and will respond sharply to your peaks and slopes. The low-pass component will remove noise beyond the cutoff frequency.

You should design your cutoff frequency so that you get clean pulses for your slopes.

Positive slopes will slow as positive pulses; negative slopes as negative pulses, and the peak will correspond to the zero-crossing between positive and negative.

This type of filter is normally implemented as a FIR filter. The number of samples for your filter will then depend on your real-time constraints, the sharpness at the cutoff frequency, and the cutoff frequency itself.