This is an achievement that requires both skill and luck. All three nations required as marches for this achievement exist in the beginning, but are often conquered quickly by their larger neighbors. Their cores will then disappear within 50-150 years.
It is recommended to read the guide for 教宗国 for a guide to start up the game. In essence, the player needs to do two things: Expand their power base in Italy so that they can field a larger army, and expand eastwards through the Balkans as soon as possible. To get a head start, it is recommended to join the Holy Roman Empire by allying and improving relations with 奥地利 as quickly as possible. Otherwise, expansion will be severely restricted until Italy leaves the empire around 1490, with the only non-Empire nations initially within reach being 普罗旺斯, 威尼斯, 那不勒斯, 匈牙利 and 法兰西, all of which will be able to field a larger army than the player initially. If the player joins the HRE, they may expand through the smaller nations in Northern Italy such as 佛罗伦萨 and 锡耶纳. Take note that as long as the Papal State is Curia Controller, they can likely excommunicate their rivals for drastically reduced aggressive expansion. In general, even with the excommunication casus belli, expansion in Northern Italy will likely have to proceed at a relatively slow pace, due to the high development of the provinces. If feasible, the player can also attack 阿拉贡 with either 法兰西's or 卡斯提尔's help in order to expand their power base in Italy.
As soon as the player has a foothold in the Balkans - most likely by taking provinces from 威尼斯 or 匈牙利 - they should expand there as quickly as possible. 波斯尼亚 and 塞尔维亚 can be annexed entirely with relatively minimal impact, and 拉古萨 can be taken over for extra trade power if the 奥斯曼 dishonor or drop their guarantee. At the earliest opportunity, the player should also beat back the 奥斯曼 for a stronger power base in the Balkans and Greece, and to generally weaken them.
Both 奥地利 and 波兰 make great initial allies; Austria for entering the HRE temporarily until the Shadow Kingdom triggers, and Poland for helping defend against and/or attack the 奥斯曼 and 匈牙利. Take note that the alliance with Poland may need to be dropped once the player is strong enough (and has other strong allies), in order to get a land connection with the Livonian and Teutonic Orders.
A good starting idea group for this achievement are the influence ideas, due to the reduction in aggressive expansion impact, which stacks with the Papal State's 7th national idea, but also due to the additional diplomatic relations slot. A military idea group will be of advantage as the second idea group, and the third should most likely be administrative ideas. The Papal State's tolerance of heathens is so low that even humanist ideas may not help, and religious ideas may not be necessary, due to the Papal State's buffs to missionary strengths.
医院骑士团 is the easiest of the marches to get - the player may either diplo-vassalize them once they are close enough, or attack them directly once they are in reach, while taking heed of 威尼斯's army. Take note that due to Papal State's lack of royal marriages, diplo-vassalization usually takes more effort. Additionally, The Knights cannot be directly released upon conquest of Rhodes, but rather the core needs to be returned and the new state diplo-vassalized or force-vassalized instead.
The Teutonic and Livonian Orders
For much of the early game, the player will likely be unable to core any of the provinces in the Baltic Sea without either a land connection to that sea, or a vassal in the area. The only expansion directions are either through the HRE - unlikely, as the player should leave it by 1490 when the Shadow Kingdom triggers - or through Genoa / Crimea / Muscovy (likely with Poland's help), or through Poland itself (with the help of Poland's enemies).
The 条顿骑士团 is usually one of the first targets of 波兰 for expansion, meaning their cores are likely to disappear in the 1500s or 1600s. Once their cores have disappeared, the player may not be able to resurrect them again. Also ensure that the Teutonic Order remains Catholic or does not hold Koningsberg, else they might form 普鲁士, which renders the player ineligible for the achievement. The Teutonic Order does not need to be independent to form Prussia, so vassalization is not enough to prevent them from forming Prussia. The player is advised to check on the duration of these cores and plan accordingly - this can be done by viewing the province and hovering the Teutonic Order flag there. This will likely need to be the second force-vassalization or vassal release. An ideal situation would be if Poland decides to not form the personal union with Lithuania, in which case the northward expansion will become much easier.
The 利沃尼亚骑士团 is more likely to survive until the later game, but may also be absorbed by 波兰, 立陶宛, 莫斯科 or even 诺夫哥罗德 or 瑞典. Once the player has the Teutonic Order as march, reaching the Livonian Order should not be overly difficult. However, since the Livonian Order initially holds no provinces of their primary culture (Prussian), they cannot be released if they ever disappear.
Jerusalem is initially in the hands of the 马穆鲁克, but it is very likely that the 奥斯曼 conquer it within the first 50 years of the game. The player essentially has two options, depending also on how the game develops, and which allies both the player and the Ottomans are able to make: If the Ottomans are weak and the player is within reach, it is best to seize the opportunity and attack, weakening them as much as possible. If that is not possible early and the Ottomans are able to expand and grow strong, the player may need to wait until they are both strong enough and have strong enough allies to beat the Ottomans back. It will be a good idea to take advantage of the Papal Controller status to call for a Crusade against the Ottomans or their allies (likely to include 突尼斯), to encourage other nations to attack them.