Are GnuPG 1 and GnuPG 2 compatible with each other?

GnuPG 1

GnuPG 1.4 ("classic") will stay for embedded and server usage, as it brings less dependencies and smaller binaries. Earlier, it was often installed as gpg, today it is more often named gpg1 (depending on distribution).

From the GnuPG 1.4 man page:

This is the standalone version of gpg. For desktop use you should consider using gpg2 from the GnuPG-2 package (On some platforms gpg2 is installed under the name gpg)

GnuPG 2

GnuPG 2.0 is a redesigned version of GnuPG -- but changes are mostly on internal level. The newer version is split into multiple modules, for example there are also modules for X.509 (used by S/MIME).

From man gpg2:

In contrast to the standalone version gpg, which is more suited for server and embedded platforms, this version is commonly installed under the name gpg2 and more targeted to the desktop as it requires several other modules to be installed.

GnuPG 2.1

An important change came with GnuPG 2.1, which combines the formerly separated public and private keyrings (pubring.gpg vs. secring.gpg) into the public keyring. This has been implemented in a manner keeping things compatible, so you can still use GnuPG 1 when GnuPG 2.1 integrated the private keyring, but changes to the private keys will not show up for the respective other implementation. From the changelog:

[...] allows co-existence of older GnuPG versions with GnuPG 2.1. However, any change to the private keys using the new gpg will not show up when using pre-2.1 versions of GnuPG and vice versa.

To directly answer your question:

Should I install both, or just one is enough? Does it matter which one I install first matter at all?

Just install both. They don't interfere anyway. Install gpg (if not installed anyway) for other applications that access it (like package managers, mail clients, ...), and gpg2 for "direct use" on the command line.

If I installed both, are both compatible with each other?

Both implement the OpenPGP protocol, so they're compatible to each other regarding data shared among them. Additionally, they're (mostly) using the same commands and options, so most of the time you could switch between them arbitrarily.

GnuPG 2.1 makes changes to the private keyring invisible to pre-GnuPG 2.1 implementations (see above in the GnuPG 2.1 section).