Trees and shrubs seeds generally don’t need warm conditions to germinate. Most of them will even need to go through a period of cold weather to break the seed dormancy. This process is called stratification. It usually occurs in nature, but if you collect the seeds you can mimic the process by placing the seeds in a cold place or in the bottom tray of your fridge.
Collect the berries and fruits from your shrubs and trees when they are fully ripen. Remove the seeds from the berries by squashing them in an old flour sieve. Separate the seeds from the flesh of the berries. If you are not going to sow them straight away, dry them off and store them in a cool place.
Sow the seeds in trays or small pots of multipurpose compost covering them with grit not with compost. Put the trays or pots outside in a cold frame or in a sheltered place covered with a sheet of glass or plastic to keep excessive rain out.
If you are not going to sow the seeds you can stratify or give them a chilling period artificially. Put the seeds in a plastic bag and mix them with moistened vermiculite or sharp sand, then place the bag in the refrigerator. The chilling period may vary from species to species but a general period of up to six weeks will be required. Check the seeds regularly and when you see any signs of germination take them out and sow them in trays or pots.
For your fist experiments you can try some of the following trees and shrubs to grow from berries: cotoneaster, Rosa rugosa, gaultheria and all sorbus.