With Telegram

For text, links, etc., paste to your Saved Messages account (pin it so it is always at the top on both devices. When you are done, delete the message (except for at least one because without any messages it gets unpinned).

If it is a file, you can upload using Attach (paper clip icon).

You can paste an image and it will be available on the other device, but I think Telegram resizes it or does something else to it. So I think to keep original image you need to upload the actual file rather than pasting an image.

So on my mobile, to save so it is accessible on my laptop, in whatever app I'm in I press "share", then I choose Telegram, then press the Saved Messages "user", and press send. Easy peasy!