Hi

 Try this.....

  Here is my page 

<apex:page standardController="Contact" extensions="FStoringCheckBoxValueExtension">
    <apex:form >
        <apex:pageBlock title="Update Contact Details">
                <apex:pageBlockButtons location="bottom">
                    <apex:commandButton action="{!savingCheckBoxValue}" value="Save"/>
                    <apex:commandButton action="{!cancel}" value="Cancel"/>
                </apex:pageBlockButtons>
                
                <apex:pageBlockSection id="NewPage" title="Select The Country">
                    <apex:inputField value="{!Contact.Country__c}">
                        <apex:actionSupport event="onchange" reRender="IndiaBlock, USBlock, ChinaBlock, AusesBlock"/>
                    </apex:inputField>
                    <!--<apex:inputField value="{!Contact.LastName}"/>-->
                  <p> Last Name: <apex:inputText value="{!lastname}"/></p>
                </apex:pageBlockSection> 
                
                <apex:outputPanel id="IndiaBlock">
                    <apex:pageBlockSection rendered="{!Contact.Country__c == 'India'}" id="SampleSendInfoBlock" title="Select Places Visited">
                        <apex:outputLabel value="Agra" for="indiacheck">
                            <apex:inputCheckbox value="{!Indiaplaceist}" id="indiacheck1">
                                 <apex:actionSupport event="onclick" action="{!getSelected}" rerender="IndiaBlock"/>
                            </apex:inputCheckbox>
                        </apex:outputLabel> 
                        
                        <apex:outputLabel value="Delhi" for="indiacheck2">   
                            <apex:inputcheckbox id="indiacheck2" value="{!Indiaplace2nd}">
                                <apex:actionSupport event="onclick" action="{!getSelected}" rerender="IndiaBlock"/>
                            </apex:inputCheckbox>
                        </apex:outputLabel>
                        
                     </apex:pageBlockSection>
                </apex:outputPanel>
                
                <apex:outputPanel id="USBlock">
                    <apex:pageBlockSection title="Select Places Visited" rendered="{!Contact.Country__c == 'America'}" id="USPageBlock">
                        <apex:outputLabel value="Newyork" for="Newyorkcheck1">  
                            <apex:inputCheckbox value="{!Newyorkplaceist}" id="Newyorkcheck1">
                                <apex:actionSupport event="onclick" action="{!getSelected}" rerender="IndiaBlock"/>
                            </apex:inputCheckbox>
                         </apex:outputLabel>
                         <apex:outputLabel value="Washington" for="Newyorkcheck2">       
                             <apex:inputCheckbox value="{!Newyorkplace2nd}">
                                 <apex:actionSupport event="onclick" action="{!getSelected}" rerender="IndiaBlock"/>
                             </apex:inputCheckbox> 
                         </apex:outputLabel>       
                    </apex:pageBlockSection> 
                </apex:outputPanel>
                
                
                
        </apex:pageBlock>
    </apex:form>
</apex:page>

 My class is...

public class FStoringCheckBoxValueExtension {
    private  Contact contact;
    public String Indiaplaceist{get;set;}
    public String lastname{get;set;}
    public String Indiaplace2nd{get;set;}
    
    
    public String Newyorkplaceist{get;set;}
    public String Newyorkplace2nd{get;set;}
   
    public String resultString{get;set;}
    public FStoringCheckBoxValueExtension(ApexPages.StandardController controller) {
        Contact contact = new Contact();
    }
    public void savingCheckBoxValue()
    {   
        
        
        //upsert acct;
        //System.debug('@@@@@@@@@@@@@@@@@@@Indiaplace@@@@@@@@@@'+Indiaplace);
         System.debug('@@@@@@@@@@@@@@@@@@@resultString @@@@@@@@@@'+resultString );
         Contact contact = new Contact();
        contact.PlacesVisisted__c = resultString ;
        contact.LastName= lastname;
        insert contact;
    }    
    public PageReference getSelected(){
        System.debug('@@@@@@@@@@@@@@@@@@@Indiaplace@@@@@@@@@@'+Indiaplaceist);
        if(Indiaplaceist == 'true'){
            if(resultString !=null){
                resultString = resultString+' '+ 'Agra'; 
            }
            else{
                resultString = 'Agra'; 
             }      
           System.debug('@@@@@@@@@@@@@@@@@@@resultStringtestIf @@@@@@@@@@'+resultString ); 
         }
         else if(Indiaplace2nd == 'true'){
              if(resultString !=null){
                  resultString = resultString+' '+'Delhi' ;
              }
              else{
                  resultString = 'Delhi' ;  
              }     
          }
         else if(Newyorkplaceist == 'true') {
             if(resultString !=null){
                resultString = resultString+' '+'Newyork' ;  
             }
             else{
                 resultString = 'Newyork' ;  
              }   
         } 
         else if(Newyorkplace2nd == 'true') {
             if(resultString !=null){
                 resultString = resultString+' '+'Washington';
             }
              else{
                  resultString = 'Washington';
               }
         }
         else{
         }   
         
        return null;
    }
}

 Did this post answers your questions if so please mark it solved so that others get benifited..

   Thanks