An antenna has a gain set by its construction. For instance, a simple wire forming a dipole will have a gain of about 2dB whereas a dish might have a gain of 20dB. The lower the antenna gain the less directional will be the transmitted electromagnetic wave. Higher gain antennas such as dishes have to be pointed in the correct direction for them to be effective but, give you a bigger signal when receiving or concentrate the power output more when transmitting.
You can't affect antenna gain by adding an amplifier.
if "for example" the Tx has a power of 20 dBm , antenna has a gain of
10 dBi & antenna amplifier "repeater" has a gain of 60 dB "like common
cellular repeaters" , what does this gain mean ? is it mean the
antenna gain will be 10+60=70 dBi ? or it mean the Tx power will be
20+60=80 dBm ? I am confused !
- The antenna gain is fixed by it's construction
- Gain is a word that describes how much something amplifies a signal.
- Tx power is 20dBm irrespective of antenna gain BUT this power is projected into a tight beam on higher gain antennas therefore it is equivalent to an isotropic antenna (transmits the power equally in all directions) fed from 20dBm increased by 10dB = 30dBm.
another example for Rx antenna , if we have receiver has a sensitivity
of -114 dBm and Rx antenna has a gain of 10 dBi & we connect it with
40 dB amplifier , should it mean the total sensetivity will be
-114-10-40= -164 dBm ?"ignoring SNR please"
It's impossible to ignore SNR - if a receiver has an input sensitivity of -114dBm then pre-amplifying the signal by 40dB will likely increase noise and signal together and this will mean exactly the same SNR and no net benefit.
if we use Tx has a power of -30 dBm at 1000 m, antenna has a gain of
10 dBi & at 1000 m distance there is a repeater has a gain of 60 dB ,
how to calculate the total gain at this situation ?
You need to draw a sketch of something here - where is the final receiver (and I don't mean the repeater). Maybe something like this but with ALL the information on: -

Antenna efficiency is generally taken as the amount of total power radiated (in all directions) divided by the electrical power put into the antenna. It is not related to antenna gain.
Antenna gain is something a little more complex. A mathematical device called an "isotropic antenna" is a theoretical antenna that emits power equally in all directions. This is the base line for a lot of comparisons between antennas. No such antenna exists but it is still a good device for making comparisons.
An antenna that bunches the emitted power up so that it produces more watts/sq meter in one direction compared to a different direction is generally specified as having gain. A dipole antenna will concentrate power like this: -

It has a gain that slightly exceeds (by 2.15dB) the 0 dBi mark on the polar diagrams. Note I used the term "dBi" to indicate gain relative to the theoretical isotropic device.
So, an antenna with "gain" must has directionality - it forces more watts per sq meter in some directions and fewer watts per sq metre in other directions.
Antenna gain works equally in transmit as it does receive - an antenna with higher gain will collect more watts and convert them to more volts in its "optimum" direction than it does in other directions.
A receiving antenna has, what is known as, an aperture (think of it as a net which it catches radiated power - the bigger the net the more power it catches).
Both efficiency and gain need to be considered together though - an antenna with significant gain may still not be a good emitter if the efficiency is poor.
Best Answer
Rule of thumb: physically larger antennas have narrower beam width for the same frequency.
If you want to make a 30 MHz monopole more directional, you can't just cover it with a piece of metal. One solution would be to get a bunch of antennas and build a linear array, where the array factor would give you the gain. This array will need to be at least several meters long. Adding a reflector behind the antenna would increase the gain a little bit, about a factor of two. Bottom line, if you want a high gain at 30 MHz, you're going to need a REALLY big antenna.