Looks like you have an a lower end, inexpensive crank. The chain rings are riveted together, that's a construction method typically used on lower end cranks. Nothing is supposed to be removed or disassembled on such cranks.
The rings moving on the splines will cause increased wear and eventually more movement. There's nothing you can really do to fix it so a replacement crank is necessary.
Obviously you need a replacement crank with the same number and size of chainrings, but it also needs to be for the correct number of sprockets on the rear wheel. Cranks also come in a range of arm lengths measured from center of axle to center of pedal axle.
The annoying thing about square taper axle bottom bracket cranks is that each model of crank requires a specific axle length. The axle you have now may not be the correct length for whichever crank you choose to replace it. Shimano provides extensive documentation that will tell you what axle length is needed but it's often easier to simply replace the bottom bracket with the crank. On that note you should also look at replacing you chain and cassette or freewheel with the crank.