Do you have some feel for what area of network-related study you're most interested in?
Under the umbrella of "Computers & Networking" there are quite a few more specific areas, like LAN, WAN, Application, Security, Wireless, Desktop, Database Administration, VoIP, Servers/Platforms, Service Administration, Cabling & Infrastructure (with a possible sub in Optical {SONET, DWDM}) ...
Then under those general categories, there are a number of subordinate skill sets that may or may not be careers in themselves.
In a large enterprise, you'll find departments for each of most of the above. In a smaller organization, you'll find "the two network guys" and they handle everything, or they handle company-critical stuff and the rest is outsourced.
Most of the above-listed have specific certifications.
For general networking in an average company, Cisco Certifications tend to be the standard for network infrastructure, even if the company is not running Cisco stuff.
Microsoft certs tend to be requested for Platform, OS, core application, programming, database , network administration, and Exchange email.
Nortel also has some excellent certs, including PBX and VoIP, real time, and general switch/router infrastructure. There mesh wireless is very popular for metro wireless applications.
Planet3 has generic (Non vendor specific) certifications for wireless CWNA, CWNP, CWSP and they are excellent programs and well-recognized
The CompTIA courses / certs are generic and cover the basics pretty well. They are less known to HR, so they tend to not show up on job postings.
I'd forget security for a while ... you can learn it, but most companies are not going to put their security in the hands of a relative newcomer. Once you get some time under your belt then you are more likely trust your judgment and skills. There is a pretty broad spectrum of security certifications; the lowest ones are "difficult."
Juniper has some classes & certs, they tend to be advanced because (except for the smaller firewalls) Juniper tends to be in or near the core or high-capacity edge.
Anyway ... (pardon the ramble) ... it would be helpful if you could narrow things down a little.
The most-traveled path (IMO) tends to be CCNA, CCNP (possibly followed by CCDA, CCDP) with a possible MCSE for admin (SQL2K and Exchange electives) and now some sort of wireless and some sort of security (PIX, ASA, Linux-based) to round out the semi-perfect all-around network kinda person.
Supplementing all of the above, it is generally helpful to have some background with *nix (Solaris, Red Hat Enterprise, Suse Enterprise, AIX, HP(s)ux) ) and some scripting knowledge (Perl, Python, VBscript, shell scripting, etc).
Finally, to round out my blathering, it's easy to get caught up in the certification stuff such that you end up working to pass the cert, not genuinely learn the background and concepts.
You will ultimately be a better engineer (if that's your intent) if you consider the certs as milestones, not goals. Don't try to squeeze by with minimal knowledge, especially of foundational topics (like sub-netting and the operational aspect of Ethernet).
All the stuff you learn at the beginning will continually re-emerge in more complex configurations later on; understanding them to some depth at the beginning will help you to figure out the workings of things you never saw before ... and resolve the problem.
Good Luck
Scott