Given infinite unencrypted and encrypted texts, can I find the algorithm? Given an infinite amount of both unencrypted and the corresponding encrypted text, would I be able to calculate the algorithm? I am relative new to cryptography and don't know if I would be able to find how the unencrypted text is encrypted... Is this possible? If so, what should I study?
summmary

In general, no.
An attacker who has lots of ciphertext+plaintext pairs may never be able to reverse-engineer an algorithm from them.
An attack may not even be able to distinguish which one of a large group of known encryption algorithms was used to generate those ciphertexts.

However, various weaknesses in some algorithms and protocols are known.
If such an algorithm was used, the attacker may find it relatively easy to rule out all but a few or perhaps only 1 of the known encryption algorithms.

details

Some people who build modern ciphers try to make their ciphers "indistinguishable from random".
Several modern ciphers (especially ones intended for use in steganography) are not currently known to be susceptible to such a distinguishing attack.

No matter how many ciphertext+plaintext pairs the attacker collects that were properly encrypted using a single one of those algorithms using a single long-term key,
there is no known practical way for that attacker to distinguish a new message encrypted using the same algorithm using the same long-term key from a freshly generated series of bits from a hardware random-number generator.
This implies that there is no known practical way for that attacker to discover which one of those algorithms was actually used.

Cryptanalysts have discovered distinguishing attacks for many vulnerable ciphers.
If an attacker collects enough ciphertext-plaintext pairs encrypted by such a vulnerable cipher, that attacker can rule out most other ciphers, narrowing down the possible algorithm to a few or perhaps only 1 known vulnerable encryption algorithm.

When an attacker has samples of both the plaintext (called a crib), and its encrypted version (ciphertext), the attacker can mount a known-plaintext attack.

The people who build modern ciphers try hard to make their ciphers immune to known-plaintext attacks.
Most modern ciphers are not currently known to be susceptible to known-plaintext attacks.

However, many historic ciphers (most famously, the Enigma cipher) can be partially or completely broken by a known-plaintext attack.

Many distinguishing attacks and known-plaintext attacks depend on the person doing the encryption using the same key for large numbers of messages.
Sometimes that number is so extremely large that these attacks are impractical.
Other times that number is large but not improactical, and cryptographers recommend periodically rekeying long before such attacks are practical (RFC4253, RFC4344, etc.).                                     

In practice, many communication protocols transmit the specific encryption protocol used, transmitted in plaintext in the header before any ciphertext, at the beginning of every message.
An attacker only needs to look at that header to know which encryption algorithm was used.

If someone encrypts enough messages with the same key using a block cipher in electronic codebook mode,
and an attacker gets enough ciphertext-plaintext pairs,
then even without knowing which particular block cipher is used,
the attacker can decrypt any block in any new encrypted message whenever a block of plaintext in the new message happens to match a block of known plaintext.

In theory, if an attacker has an "infinite amount" of computing power,
that attacker try every possible encryption key in every known encryption algorithm and see if it "works".
Such a brute-force attack can in theory recover the key for almost all known encryption algorithms.
(The one-time pad and closely related ciphers are completely immune to a brute-force attack).
Fortunately for people who are not attackers, the amount of work required for such a brute-force attack usually makes them impractical.