Plugging in a Power Spike/Surge protector to a Voltage Converter

Can I connect a Universal Travel adapter( I own this) to a power spike and connect appliances?

You can connect your universal power adaptor directly into a US outlet. You generally don't need a surge protector. I don't know what "power spike" is. If you mean a "power strip" than, yes, that's ok too provided you can make the connection safely. Many of the cheaper adapters don't fit well and may fall out of the wall socket.

Is it safe?

It's safe if you stay within the safe limits of the device. That means

1. All the appliances that you connect MUST have a universal power supply and accept 110 V. That is often the case for chargers (phones, laptops, tablets, etc) but you need the check your clipper and any other device. Look for a voltage rating of "100V-240V". It if says "200-240V" or similar, you can't use it.
2. The total current drawn by all devices must NOT exceed 6 Ampere. That means the total power of all device should not exceed 500W or so. For low power devices, that's typically not a problem.

for things like hair dryer ...

Keep in mind that most high-power appliances are NOT dual voltage. In this case, you can't use a travel adaptor but you MUST use a step up transformer. These are heavy and expensive: it may be cheaper and more convenient to buy a new hair dryer.

When we moved to the US, we bought to step-up transformers for the entire household but mostly replaced all appliances.

Laptops nowadays always have universal power supplies. They also usually have detachable power cables. So my suggestion is to just get a US power cable for it. Just ask around, someone at your job or school (depending on why you are moving) might just have one lying around.

Mobile phone power supplies are nowadays almost always universal as well, so you could get an adaptor. But I think that just buying one in the US will be cheaper.

Hair driers are typically not suited for multiple voltages. So you would not only need a plug adaptor, but also a step down transformer. I would suggest you just don't bring one, and buy one in the US. They are not expensive either.

Most (but not all) modern electronics have a wide input voltage range and can operate off the mains voltages used anywhere in the world without any changes. Some equipment (including older electronics and some non-electronic equipment) can be used on multiple voltage ranges but requires reconfiguration, this may be done either by a switch, moving an insert, or in some cases may require internal changes. Some equipment will only operate on a single narrow voltage range voltage.

Equipment should have a ratings plate or label indicating the acceptable range of nominal input voltages. If a single wide range (e.g. 100-240V) is indicated then it can be used anywhere without changes, if two separate ranges are indicated then it will generally require reconfiguration.

Single voltage equipment would require a step-up transformer, this will be bulky and heavy, especially for high-power equipment. It's unlikely to be worth it for things like hairdriers, just leave those behind if they aren't voltage switchable and get new ones in the USA.

I recommend avoiding universal adapters for long term use. They usually lose the ground, are very bulky and often of questionable production quality. The preferred option IMO when bringing in an appliance for long term use is to replace the plug or lead. If that is not possible then I would suggest either looking for good-quality specific adapters or replacing the plug on a power strip.

Using a 240V power strip on a 120V supply or a step up transformer is fine as long as you respect current ratings, it's the other way round where you can have problems.