As an early poster said -- he is plenty old enough for consequences at home based on behavior at school. Very young children do not yet make the connection between something they did hours ago and a discipline that takes place later, but he is old enough to connect it.
Ensure that you know each day where his "stick" is (red, yellow, green)> Make it the very first thing you and he do when he gets in the door: You know his color; you ask what happened (or what GOOD thing happened that he stayed on green today); and you institute the appropriate consequence, that moment.
Be certain to reward him for green, maybe along the lines of "Two green days in a row is an extra 15 minutes of TV" or whatever, and "Three green days in a row is X, one whole five-day week on green is [some huge activity he really wants like an hour at the arcade on Saturday]." Praise him a lot, and be sure to praise him when he does things at home too. A lot of parents of young kids don't think to praise them for stuff that we adults feel should just get done, like remembering to pick up clothing and put it in the hamper, but young kids do need the reinforcement of praise, so do it even if you feel you're "overdoing" it.
For yellow, talk with him briefly (not at length! You'll lose his attention) about what specifically he will do differently tomorrow if he was on yellow today. Remind him of that specific action at the start of the next day.
Have a real consequence for red. Take away something he really values. Frankly it needs to be something he will truly feel losing, even if it makes him cry to lose it at first. He clearly does not give a darn about losing recess time because it's compensated for by the negative attention and giggles etc. that he's getting when he acts up. Figure out his "currency," the thing that means enough to him that he will work to keep it--and the work means staying on green. It might be TV time or computer time or a specific toy (or even several toys removed all at once if he would just shrug and move on to other toys when you take away toy X). You may have to really sit down and think about what the currency is.
Don't spring any of this on him. Sit down with him when he is calm and has been doing OK at home that day, and let him make a chart together with you -- a chart that reflects the school color scheme. Explain calmly that what he does at school, you always know about at home, and what he does at school is not separate from his life at home. He needs to get that; kids need to understand that their school "life" is not something they shed when they walk in the door of their house. Emphasize that there will be rewards for green and consequences for red, and yellow is not acceptable -- in fact, if he spends more than, for instance, one day in a row on yellow, I'd have a consequence but not as severe as for a red day.
It sounds detailed and maybe harsh, but he might need the structure. Kids need to know what to expect each day and this way, he would know what to expect based on the color he gets for that day. Maybe run the idea by the teacher, too.
It's interesting to see folks on here leaping to "he needs to be in a different school" and "he's a kinesthetic learner" and "you should get him evaluated for ADD and ADHD". While yes, some kids do learn better if they are moving around, you said nothing in your post to indicate that's the issue yet, and nothing to indicate that he has other signs of ADD or ADHD. What you describe sounds like basic, natural immaturity that many a child goes through. A structure for rewards and consequences that links his school-day behavior to what happens at home could help. You would need to stick to it for quite a while, I think, but it's a possibility. If he shows other signs over time of needing a diagnosis of some sort, yes, absolutely have him checked, but what you describe sounds simply like normal adjustment problems transitioning from K to first grade.