As long as they've ripened, seeds can be gathered from most trees, shrubs, annuals, biennials, bulbous plants, ornamental grasses, vegetables and herbs.
If they're collected, dried and stored carefully, they'll remain viable for up to five years. 
Since seeds come in many different types of packaging (fruits, berries, capsules, catkins, cones, exploding seedheads, nuts, pods and winged seeds), there are slight variations in the ways to collect them. However, below I've tried to give a general summary of best practise. If you need more details visit the RHS and Gardeners' World websites, which were both used for this post. [2,3]
To collect seeds you'll need the following:
Paper envelope (free inside the seed bank)
1. Wait for seeds to ripen on the plant before collecting them. This will be when the 'packaging' (seedheads/berries/nuts/etc) changes colour from green to brown, black or red.
Exploding seedheads will need to be removed on their stems as they turn brown, and placed in a labelled paper bag
Nuts should be collected around the time they would naturally fall by hand-picking
Fleshy fruits and berries should be mashed in a fine sieve, and the seeds separated by rinsing the pulp away in cold water
2. Ensure seeds are dry
Try to collect ripe seed on a dry day
If seeds are damp, lay them out to dry on a greenhouse bench, warm windowsill or in an airing cupboard. Gently crush pods and capsules to release the seed if they don’t open when dry
Seeds collected from fruit should be left to dry for a few days on paper towels
3. Ensure seeds are clean
After extracting the seed, clean off any surrounding material to reduce the risks of rot and the seed damping off. Chaff can harbour moulds, pests and diseases
4. Funnel the seeds into paper envelopes (you can grab some from the seed bank), not plastic bags*, and write full details of the plant (name and colour) on the envelope, including the date.
Storing seeds Some seeds (e.g. hellebore) are best sown immediately as their viability reduces with storage. However, many can be stored safely until sowing.
Place envelopes of seeds the correct alphabetical container inside the seed bank
*Certain seed must not be allowed to dry out as they cannot then take up water necessary for germination. Examples are walnuts, oaks and magnolias. These seeds can be stored in a plastic bag of damp vermiculite, sand, or a mix of moist coir and sand for several months
I hope you found this helpful! Happy seed gathering :)
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