You don't necessarily have a problem or need an upgrade. It is also not usually dangerous, more just a nuisance. Truthfully you would never know if your lights didn't dim momentarily.
Most motors will draw between 4 and 6 times rated current during starting. Most equipment will have some sort of inrush current when first turned on. The TV will degauss its self and the CPU's power supply will have inrush.
If these are on the same circuit as your lights then you will get a slight dim. This is because the momentary high current will produce a momentary voltage drop on the circuit. After this inrush is over the momentary drop ends and your lights brighten.
If this is a single circuit you can alleviate the problem by moving the loads off of a circuit that feeds the lights. If this cant be done by just moving to a different closeby receptacle then you may have to rewire to get the lights and receptacles on different circuits. If this is a branch circuit problem like this an upgraded service won't solve the problem because it does not address the cause of the problem.
If the lights dim continually when the load is on then you just have some voltage drop problems. This is due to long circuit runs and can be mitigated by reducing the load on that circuit or rewiring with larger conductors to reduce the voltage drop. An upgraded service won't solve this problem.
If this is something that occurs on your whole house then you still may not have a problem and even so whats actually causing it may be totally out of your control.
Lots of times the starting inrush on a heat pump will dim the whole house. This does not mean the service is inadequate. The major voltage drop thats being experienced is most likely found in the power company conductors between your service and the transformer. The power company follows different rules than an electrician will and the wire between the transformer and the service will usually be smaller than the service entrance conductors of your service. Smaller conductors mean a greater voltage drop and a more noticeable flicker. You can probably talk to the power company enough to get them to change this out.
This can also be caused by the impedance of the transformer, this is totally out of your control as well and a new service wouldn't help you a bit in this situation. All transformers have an impedance rating. Without getting too technical the transformers impedance is directly related to the transformers voltage regulation. Most power company transformers have a fairly high impedance, meaning that when the load is increased the voltage will droop by a higher amount than a lower impedance transformer.
You may have a very hard time getting them to change this. They usually won't change one out for relatively minor problems like flicker. It is also a safety concern to have a transformer with a very low impedance. The lower the impedance you have, the higher available fault current you will have. Since equipment is only rated to interrupt a certain fault current you will find that the power company errs greatly on the side of safety when it comes to available fault currents. You don't want them to give you a low impedance transformer just to eliminate flicker and have your breaker box blow up because the fault current rating of the breakers has been exceeded.
Also I bet you are using incandescent bulbs??
If so, incandescent bulbs are VERY susceptable to even the slightest variation in voltage. A drop of just a couple volts is noticeable to the human eye. When the drop occurs very quickly due to motor inrush the resulting quick dip in lumens is easily noticed. What you probably dont notice is the difference in light between midnight and the middle of the day. At night your voltage is usually a bit higher because the rest of the world is not using power, however during the day its usually a few volts lower. The intensity of the light bulb will changed during all this, it just happens so slowly that you don't notice.
Try switching to compact fluorescent lights. They tolerate voltage variation much better than incandescent. You may not notice the flicker at all if you change! You will also be doing yourself a favor and save on your power bill and will do the world a favor because they require less power and therefore less fuel. They generally last a lot longer too.
The only way I would say you needed to upgrade your service would be if you turned on the TV and the lights in the WHOLE house dimmed and STAYED dim, not just a flicker. Even then the voltage drop concern would likely be solved by larger conductors from the transformer to your service.
I am an Electrical Engineer for a large chemical company and specialize in power distribution. I also have previous education and experience as an Industrial and Residential/Commercial Electrician.