How to set record type of records while saving the records?

SUMMER '18 UPDATE

You can now use the getRecordTypeInfosByDeveloperName() to get a Record Type Id by Developer Name without spending a SOQL query. This should be the preferred approach going forward instead of using the old getRecordTypeInfosByName() method as suggested in the original answer.

Use this approach instead of the old suggestion below, unless you are really trying to get the Record Type Id by RECORD_TYPE_NAME (Label). The rest of the answer still applies:

Id recordTypeId = Schema.SObjectType.OBJECT_NAME.getRecordTypeInfosByDeveloperName()
                      .get('RECORD_TYPE_DEVELOPER_NAME').getRecordTypeId();

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You can obtain a Record Type in code in the following way:

Id recordTypeId = Schema.SObjectType.OBJECT_NAME.getRecordTypeInfosByName()
                  .get('RECORD_TYPE_NAME').getRecordTypeId();

Just replace the OBJECT_NAME with your object (e.g - Account), and the RECORD_TYPE_NAME with the record type name for that object.

In order to assign this Record Type to a specific record you can use the RecordTypeId field. Here's an example for Account:

Account acc = new Account(Name='Test Account', RecordTypeId = recordTypeId);

or through a property like this:

acc.RecordTypeId = recordTypeId;

UPDATE

In your case you can obtain the recordTypeId like this:

Id recordTypeId = Schema.SObjectType.Grant_Report__c.getRecordTypeInfosByName()
                  .get('Matching Fund Cash').getRecordTypeId();

And then just assign this recordTypeId to RecordTypeId field of the records in your FOR loops that you need to update. I'm assuming that's in the saveDoc() method:

 Public Pagereference savedoc(){
    Id recordTypeId = Schema.SObjectType.Grant_Report__c.getRecordTypeInfosByName()
                  .get('Matching Fund Cash').getRecordTypeId();

    List<Grant_Report__c> allGrantReports = new List<Grant_Report__c>();

    for(Grant_Report__c gr: grCashObjLst){
             if(gr.id==null)
             {
                  gr.Grant_Approval__c= ids;                   
             }
             gr.RecordTypeId = recordTypeId;//assign record type
             allGrantReports.add(gr);
     }
     for(Grant_Report__c gr: grInKindObjLst){
             if(gr.id==null)
             {
                gr.Grant_Approval__c= ids;
             }
             gr.RecordTypeId = recordTypeId;//assign record type
             allGrantReports.add(gr);
     }
     for(Grant_Report__c gr: grExpenseObjLst){
             if(gr.id==null)
             {
                gr.Grant_Approval__c= ids;
             }
             gr.RecordTypeId = recordTypeId;//assign record type
             allGrantReports.add(gr);                 
     }
     grExpenseObjLst=null;
     try{ 
           upsert gaObj;          
           //saved you 2 DML statements here by joining 3 lists to 1
           upsert allGrantReports ;          
     }catch(System.DMLException e) {
            ApexPages.addMessages(e);
            return null;
       }  
    return NULL;
  }

UPDATE 2

NOTE: No need to use this approach after Summer '18 update

In case you want to get Record Type by developer name, you'll have to use a DML statement:

Id recordTypeId = [select Id from RecordType where DeveloperName = 'Matching_Fund_Cash' AND sObjectType = 'Grant_Report__c' limit 1].Id;