I'll try to answer your question, sorry if it doesn't satisfy you. I'm answering on the Ottoman side

how spies traveled to other countries without getting caught

Disguises/blending and using profession as cover are frequently used by spies to infiltrate enemy countries. Many if not most Ottoman spies are people that could easily blend in the societies they're send in. For example, Constantino the spy that was sent to Trapani was a Greek Muslim convert who donned Christian attire to enter the city; Baron de la fage the infamous Ottoman spy was a Frenchman; a certain soldier that was sent to spy on Salerno can speak Spanish and pretended to be a Spaniard; and on 1560 the Ottomans sent a spy to Venice in disguise of a merchant.

Another way is to use profession to provide mobility to other countries. In the early modern era, spying was more often than not a secondary job, with the primary job providing opportunities, access, and mobility needed to obtain information. Merchant, diplomat, corsair, and clergyman were some of the primary profession of many spies. For example, Milanese born pro-French archbishop of Ragusa Filippo Trivulzio frequently provided information to the Ottomans and the French; French ambassador Guillaume Pellicier furnished the Ottomans intelligence from Venice; Yunus Bey the Ottoman ambassador to Venice successfully obtained maps of Istria and Dalmatia; and the corsair Uluç Ali maintained a network of spies he managed himself.

Another way is to use fifth columns to furnish information. Moriscos for example provided a lot of information traffic for the Ottomans. For example In 1552, a morisco, Alonso de Ayala ran way from the fortress of Djerba to the Ottoman side. He would later settle in Istanbul and provided the Ottomans information he received from his brothers in the fortress of la Goletta. Neapolitans were also a big part of Ottoman espionage in the Italian Peninsula. The Ottomans maintained a network of spies and collaborators in cities like Salerno and Syracuse. Lastly, Jewish communities were also frequently included in Ottoman intelligence networks

Of course, this doesn't meant that these all were foolproof way to spy. Spies will often get caught, ambassadors blocked from entering a country, and people of professions got their cover blown up. However, the Ottoman spy network was quite extensive that a caught spy could be quickly replaced or information obtained elsewhere

what kind of information they collected

All kinds of information! Troop and fleet movement plans, numbers of soldiers garrisoning a city, the condition of a fortification, and political conditions were some of the information that an Ottoman spymaster might request from his spies. Constantino the spy was tasked in 1561 to gather information regarding Syracuse's walls, guns, and garrisons. The soldier sent to Salerno was given a comprehensive list of things he should know, from public opinion of the Habsburgs to plans made by the Habsburg navy.

how did they make sure the information they gathered wouldn't become obsolete once they arrived at their own country?

In the case of the Ottomans and Venice, the Venetians maintained an effective postal service between Venice and Istanbul so naturally the Ottomans, as well as other diplomats such as the French ambassador had to rely on the Venetians, even at the risk of their letters being secretly opened and read. To counter this, Ottoman spies used fake names and without revealing true intentions. Other than that, there are other methods to send information quickly. From sea, the Ottomans sent small ships, kalyete, mostly captained by a corsair, to quickly send important letters. When the communication via sea proved impossible, couriers were sent via land as well, until the nearest port. Sometimes, in order to assure that a message reached the navy, two couriers were sent, one from the land and one from the sea. For communications within the empire, the Ottomans maintained an impressive network of post-stations (menzilhane), providing supplies and horses to ensure that the messengers arrived quickly

However, from what i've read outdated information was less of a concern compared to a manipulated one. Generally speaking, it seems that most of the information sent by Ottoman spies arrived quickly. News of the Christian Armada was reaching Uluç Ali with letters from Sicily, and an Ottoman agent returned from Spain and Flanders with the news of Habsburg military preparations. Additionally, the Ottomans maintained an extensive spy networks that allowed them to diversify the channels of information and effectively compare and contrast the news that reached them. Aside from spies, there were corsairs in the Adriatic and Mediterranean, provincial governors in border cities, the vassal state of Ragusa, and when all these failed, there were always European diplomats in Constantinople who could share information, including even the Venetian bailo. Even though under house arrest, he was still corresponding with Venice and so could be pressured to reveal information - indeed, the Venetians were actually one of the biggest source of information regarding Habsburg activities in the Ottoman Empire

Sources:

Sultanın Casusları, 16. Yüzyılda İstihbarat, Sabotaj ve Rüşvet Ağları by Emrah Safa Gürkan

Espionage in the 16th Century Mediterranean Secret Diplomacy, Mediterranean Go-Betweens and the Ottoman-Habsburg Rivalry by Emrah Safa Gürkan

Ideology, and Limits of Imperial Policy: Ottoman Grand Strategy in the Context of Ottoman-Habsburg Rivalry by Gabor Agoston

For more Ottoman espionage, see my previous posts here and here