You will need a suitable container when you first collect the seed. Hessian or open mesh bags are good or open baskets for large seeds like acorns. If seeds cannot breathe they may become damp and overheated which reduces their chances of successful germination, and the seeds may even go mouldy.
In general, avoid polythene or plastic bags which do not allow air to circulate, but these may be used for berries which can be kept in open polythene bags for up to a month before stratification.
It is sensible to get the seeds into the seedbed, or start stratifying, as soon as possible after collecting them. However, if seed has to be kept temporarily, leave it in a suitable container in a cool dark place - a weatherproof garden shed is fine.
Scots pine cones, alder cones and birch catkins may be stored entire, and the seed shaken out immediately prior to sowing.