How do I connect multiple water-filled syringes?

If the pressure is not too high you can use something like the Luer system there are plenty of options for connectors and valves etc.. I have found parts up to 1050 PSI. But most stuff can't handle more than 50 PSI.

You can get them from all kinds of medical supply sites and even sites like ebay and amazon. I have found this site after a quick search for Luers and they seem to have all kinds of switches and connectors for Luer syringes. With this system you would connect all your tubing and valves (maybe even with a glue) and then screw the syringes on. This makes filling the system and swiching parts pretty easy. These parts are also all made from plastic so they are really cheap. I have found them to be pretty leak proof as long as you don't use something oil or too high pressure (remember it's all plastic). If you want you can always put glue on the connectors to seal them even better.

If you need to switch the flow between the syringes use something like this 3-way Stopcock I found on amazon or this high pressure (1050 PSI) one i found on qosina.com.

If you need something more durable you might not want to use syringes at all.

I don't know what exactly you want to accomplish with your setup but if you are looking for the syringes to level off or something similar you might not get good results because of the high friction of syringes. So keep in mind that movement of the syringes will probably be pretty jumpy and not at all smooth.

Another option would be to use a proper hydraulic system, but that would be significantly more expensive (at least 20x). So if cost is your main consideration and the limits of syringes are fine you should look for Luer.

One can use aquarium tubing fittings to match the tubing diameter you are using. These fittings can be found in tropical fish retail locations, as well as online.

Another source of similar fittings is the local hardware store. Our area is populated with Ace Hardware stores and I've purchased more than a few plastic barbed tubing fittings. In your application, you would be seeking tee fittings.

Some sprinkler systems use similarly barbed fittings and can be found at the big box stores and/or sprinkler companies. These are less likely to be the correct choice, as the diameter usually is 1/4" or greater.

One source for a smaller fitting (1/16") is MSC Direct. Image below courtesy MSC Direct.