"I have heard that some compressors do not need start kits."

It depends on circuit length from the main panel to the unit, how fast the unit 'bleeds down' when off, and a few other things.
If it is 9 years old adding one now is probably not warranted.

A hard start kit is another start capacitor and a sensor to remove it from the circuit (since it cannot sense motor speed).
It increase the starting torque available, and usually reduces the length of the starting current draw (the motor comes up to speed faster).

A hard start kit will not resolve the issue in your particular situation. I think you did change the start capacitor and start relay, which should help. If your unit doesn't have a start relay and start capacitor, by all means, add the kit, it won't hurt and it might minimize the nuisance. However, on a unit that size, I'm sure it has a start capacitor and relay, which is what a hard start kit is.
I do think the momentary dimming of lights during the initial surge when the compressor starts is likely what Brickeye mentioned, the distance from the main panel may be the culprit. You might also have some other high draw components on that circuit such as an electric hot water heater or stove that may be operating at the same time the unit starts. In that case, you may not see the lights dim every time the unit starts. One thing you could try is first is making sure your grounds and connections are all clean and tight, even at the panel. Follow all rules concerning high voltage while working this issue. Make sure you know what you are doing at the panel and also around connections at the unit (I think you do but must mention it). If the momentary dimming is a nuisance, you could move the breaker and associated ground to the other side of the box. This of course may make lights dim in a different part of the house.

When you say the start relay are you talking about the start contactor? That is what I changed and as far as a relay, I do not see one in the unit. This is a Carrier Weathermaker 13 seer 4 ton R134A. The lights dimming is really no big deal to me but I just don't want the compressor to burn up. I was looking at the manual on the condenser and it showed a hard start kit as an option but I don't have it,,, I just need this thing to last if you know what I mean !!

I'm guessing from your description, you changed the contactor relay which gets energized by the thermostat to power up the unit. The start relay is smaller and would be connected to a start capacitor which would look much like the hard start kit capacitor you have. If the kit is optional and you are sure there is no starting circuit already in the unit, go ahead and put that one in.
You are being proactive and that is great. I just want to throw some sand into the suntan lotion by mentioning it may not stop the lights from dimming but it might do it to a point you that is not as noticeable. It will also help to take off some of the burden on the compressor when starting which of course is when a lot of wear and tear happens.
I still service units that are well over 30 years old so although I can't say yours will benefit the same, barring other issues, it very well could.
One other thing you may want to do is, make certain the coils are clean and there are no obstructions around the condensing unit to impede air flow. You should still check all the connections and make sure they are clean and tight. Heat, under voltages and spikes in time, do kill compressor and fan windings.

Here is some advice; stop messing with your unit before you get hurt or hurt someone else.

Hey Heatseeker,

Here is some advice for YOU. Go back to charm school and learn some manners. You might find that people treat you better.