/u/DeSoulis went into the political side so I'll go into the strategic bits.
Taiwan looks small and diminiuitive on maps, especially when compared to the juggernaut of the People's Republic of China.
But one of the main reasons the nationalists moved to Taiwan was because of strategic security the location provided. The PRC navy at the time had almost zero open sea naval capability - much of it was river and coastal boats.
The other important thing to remember, amphibious landings are a highly complex combined arms operation that few nations, even today, can pull off. Opposed amphibious landings, that is, landing on a defended beach under fire, is even more difficult and inherently an incredibly risky military maneuver.
The PRC naval forces in the 1950s and 1960s lacked two major things that they needed for any theoretical scenario that pitted them against Taiwan.
1) Lack of heavy duty warships
What cruisers and missile frigates the PRC did have was either bought direct from the USSR or received massive amounts of help in construction and development by the USSR. But they did not have nearly enough for the ideal mission in the minds of PRC military planners of the time, a naval blockade of Taiwan to force capitulation.
They also did not have nearly enough firepower concentrated in their cruisers and escort ships to provide support fire in the case of the less ideal mission, a full out amphibious invasion on Taiwan.
This is not even considering the fact that the PRC navy might have to face off against US navy ships if they did invade Taiwan in the 1950s and 1960s.
2) Lack of amphibious landers/transports in general
The PRC severely lacked amphibious assault vehicles that could be used to send troops into an opposed landing. Even with ideal weather conditions as well as lots of air and naval support, World War 2 showed that opposed landings in modern warfare were brutal not just in combat but in logistics as well.
They were not a simple operation to pull off.
And the casualties were often high very high.
People often think have an image of PRC military doctrine as just sending waves after waves of men in near suicidal attacks until they overwhelm the enemy. This is a stereotype and commanders often considered combat casualties when deciding what to do. That being said, taking objectives 'at all costs' was also valid.
But in a Taiwan invasion scenario, they don't even have the luxury of considering throwing dead bodies at the problem. Any operation against Taiwan involved a massive amount of military intelligence and a command structure able to digest and act on it in a reasonable amount of time, supplying naval and land forces far away from PRC supply lines, round the clock air operations, and communications between thousands of units, 3 different command structures, etc.
The sheer logistics of amphibious landings are often overlooked but in reality, they are incredibly monumental operations when conducted in scale and only a select few military forces have ever been able to properly conduct them throughout history. It's never as simple as 'let's land 10,000 troops here'.
The logistics here were further compounded by the fact that much of the Taiwanese coastline is not suitable for a large amphibious landing. This problem is further compounded by the fact that the Taiwanese tide is considered very difficult to work with in terms of a landing.
Anyways, because of the limited number of transports (Chinese military planners expected that in case of an actual invasion of Taiwan, they would likely need to requisition every civilian boat they could to supplement military vessels), any theoretical landing operation was very risky.
Especially if US air assets/naval assets intervened.
Taiwan's interdiction ability was limited but US forces would be more than a match for a force steaming towards Taiwan.
Chinese military planners worried that staging areas in Nanjing and Guangdong would be subject to US bombing while Chinese transports would be sunk with their landing troops before they could even reach Taiwan, resulting in tens of thousands of casualties before even reaching Taiwanese soil.
These two issues were never really 'fixed'.
While PRC naval forces continued to expand and develop, they never had enough resources where they could comfortably assure officials of the success of a landing scenario, which meant in case of a Taiwanese conflict, the ideal was to enforce a blockade of Taiwan in the ocean while keeping the US out of the conflict in the political arena.
The PRC instead invested heavily into missile and SAM technology that allowed them to increase the amount of firepower they could bring down on Taiwan land and airspace.
This arguably helped a great deal, especially the SAMs as it would help them in the fight for air superiority in and along the invasion corridor, which would have been essential to keep casualties of troops and transports en route to a minimum.
But more importantly, long range missiles allowed the PRC to threaten Taiwan without having to risk a blockade or invasion.
Taiwan's greatest asset strategically was the tiny strip of water between it and the mainland.
If you have any more questions I'd be happy to answer them to the best of my ability. Cheers!