So, ive got to remove last years seedlings by the end of june, in proper pots(fortunately, finding the pots, and filling up them with soil is the easy part). The hard part, is removing the seedlings from the plastic trays, into their new home. Any idea what to use and how to do it carefully?

Grab them with several layers of newspaper.
If there to tiny to grab, they could probably go another year, but if you insist, a big pair of tweezers will work.
A picture or two would help, and be nice to see 😀🕵️👍👍
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Seedlings are too small to remove with newspaper....... i use tweezers to handle the bigger stuff.

I'd let the the smaller ones get bigger.
I know 😮 ! Use a couple of teaspoons ?
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Yes, let them grow up and fill the space they are currently in.

Here is what I do for seedling removal. Typically I will start seeds in 4 or 6 inch pots (10/15cm)... to remove them, I try to pop out the whole group as one unit, to the extent the soil has compacted a little and they have rooted through it, holding it together. Usually I can push up on the bottom of the container and the soil will move up as a unit. Once the contents of the container are out on a plate (looking like before but minus the container) I will take the point of a chopstick and carve out soil around the edge that the plants haven't rooted into yet. Places around the edge where there are gaps, soil that is not being used by the plants.

Then if necessary I will push the chopstick in between the two biggest plants, or on the inside of the plant which will be easiest to start with (ie. the one closest to the edge). I put it in vertically down to the bottom, gradually, avoiding roots I might encounter, then gently move the top of the chopstick sideways in the right direction to encourage the soil to break along the line I have chosen. Lots of discretion here, you want minimum rippage and ideally each seedling will end up still rooted in a little ball of soil at the end. I do not bare-root the seedlings on purpose, but sometimes the soil falls away.

At the end I handle and pot up the seedlings mostly by grabbing and handling the root ball, not so much the plant itself. In the best case scenario, every seedling goes into a new pot with its roots undisturbed in original soil, and then gets filled in around the edges with new soil.

As for tools, tweezers work for spiny cacti. I like to use chopsticks. But mostly I try to use my hands if possible to handle the soil more than the plant, if that makes sense. One reason why I like young seedlings to root reasonably well into their starter pot before I even consider moving them along.

Id place them in pots, but i cannot find ziplock(and lots of it). But bear in mind, this is a photo from march, i intend to deal with the seedlings at the end of june.

I do not use baggies. I reuse clear plastic containers as lids (cut down to size if necessary) and leave a flat piece of clear plastic over the top of the pot. Or I might use clear plastic wrap and tape that around the edges.

Some examples here... with no lid (golden barrels there), a hard plastic lid, or plastic wrap (which I use for the smallest seeds).

Well, cant say the tray method hasnt worked(result speaks for itself).

Thanks for the detailed explanations & pics, Baja.

Just to be clear- its not my first time handing seedling- just first time removing them from a plastic tray - tried to germinate them in some kind of germination tray with a lid , but the soil was not very good, and germination was medium at best.
The tiny cups were easier to cut and the soil to split.

I had a friend who used to do missionary trips to Haiti as part of a dental group. They had hard plastic dental kits that could be thrown away and did not need to be sterilized. I got one of those, and it has been amazing for handling seedlings. But over the years I have broken, lost, and used it so much. I have never been able to find another identical one, but you might want to look into that.