--I'm supposing this is your old table containing ALL records of visits
create table old
(
idNum int,
visitDate datetime
)
GO

--Lets fill it out with the sample data you provided
insert into old values(1,'1/1/2011'),(1,'1/2/2011'),
(1,'1/5/2011'),(1,'1/17/2011'),(1,'1/18/2011'),
(1,'1/20/2011'),(1,'1/24/2011'),(1,'1/29/2011')
GO

--Let this be your NEW table in which we're only inserting FILTERED DATA
create table new
(
idNum int,
visitDate datetime
)
GO

--Lets insert the very first visit date since it's surely an 'independent date' as you said
insert into new
select idNum, MIN(visitDate) from old
group by idNum



--Creating some variables we need
DECLARE @idNUM int
DECLARE @visitDate datetime
Declare @lastIndependentDate datetime

--Now lets run through the data in the OLD table and only insert in NEW the rows those we need
DECLARE db_cursor CURSOR FOR 
SELECT idNUM, visitDate 
FROM old
order by visitDate 

OPEN db_cursor  
FETCH NEXT FROM db_cursor INTO @idNUM, @visitDate

WHILE @@FETCH_STATUS = 0  
BEGIN  

	 Select @lastIndependentDate=MAX(visitDate) FROM new WHERE idNUM=@idNUM
       if(DATEDIFF(DAY,@lastIndependentDate,@visitDate)>10) 
       insert into new values (@idNUM,@visitDate)
       FETCH NEXT FROM db_cursor INTO @idNUM, @visitDate
END  

CLOSE db_cursor  
DEALLOCATE db_cursor

--WHAT you should do is to firstly empty your NEW table completly and execute the folowwing :

--insert the very first visit date since it's surely an 'independent date' as you said
insert into new
select idNum, MIN(visitDate) from old
group by idNum



--Creating some variables we need
DECLARE @idNUM int
DECLARE @visitDate datetime
Declare @lastIndependentDate datetime

--Google what a T-SQL Cursor is and what it's good for then..
--here's the declaration of the cursor
DECLARE db_cursor CURSOR FOR --this cursor's called db_cursor (you can name it whatever you want)
--it's a cursor on the following query result
SELECT idNUM, visitDate --we select the idNUMs and visit dates of people in your database
FROM old
order by visitDate --ad we order them ascendently on the dates column so as to easilly filter dates we don't want ;)

OPEN db_cursor  --open the cursor
FETCH NEXT FROM db_cursor INTO @idNUM, @visitDate --put the query result's current row's idNUM in the variable @idNUM and visitDate in the variable @visitDate

WHILE @@FETCH_STATUS = 0  --while there are more rows to go through
BEGIN  
--put the last 'independent date' in the variable @lastIndependentDate
--last independent date is surelly the maximum date in the NEW table since we ordered dates ascendently
--in the query before  ;)
	 Select @lastIndependentDate=MAX(visitDate) FROM new WHERE idNUM=@idNUM
	 --now check wether the current date gotten from the old table is 10 days bigger than the last independent date
       if(DATEDIFF(DAY,@lastIndependentDate,@visitDate)>10) 
       insert into new values (@idNUM,@visitDate) --if it's the case insert it in the new table,
       FETCH NEXT FROM db_cursor INTO @idNUM, @visitDate -- else don't do anything and continue on with the next row
END  
--once everything is done : 
CLOSE db_cursor  --close the cursor
DEALLOCATE db_cursor --free resources allocated by it

with cte1 as -- attaches the next visit date to every visit date
(
   select 
      a.visitDate,
      min(b.visitDate) nextDate
   from 
      old a left join old b 
         on b.visitDate >= dateadd(dd,10,a.visitDate)
   group by
      a.visitDate
),
cte2 as -- traverse the chain using the next visit date.
(
   select top 1 
      * 
   from 
      cte1 
   order by 
      visitDate
   union all
   select 
      a.visitDate,
      a.NextDate 
   from 
      cte1 a 
      inner join cte2 b 
         on b.NextDate = a.visitDate
)
select visitDate 
from cte2 
order by visitDate;