How can I see all active connections?

You are looking for the netstat command. This command should provide what you're looking for:

netstat -a

if you would also like to see what programs are using the specified ports you can use:

netstat -b

to use the netstat program:

• Go to the start menu (or press Win + r and skip to step 3)
• If on XP, click "Run", If on vista or later, search for cmd in the search box and skip to step 4.
• type cmd
• after cmd opens, type netstat -a
• a list of all open connections with their ports will be displayed

more info about netstat:

C:\Documents and Settings\Administrator>netstat /?

Displays protocol statistics and current TCP/IP network connections.

NETSTAT [-a] [-b] [-e] [-n] [-o] [-p proto] [-r] [-s] [-v] [interval]

-a          Displays all connections and listening ports.
-b          Displays the executable involved in creating each connection
            or
            listening port. In some cases well-known executables host
            multiple independent components, and in these cases the
            sequence of components involved in creating the connection
            or listening port is displayed. In this case the executable
            name is in [] at the bottom, on top is the component it 
            called,
            and so forth until TCP/IP was reached. Note that this option
            can be time-consuming and will fail unless you have 
            sufficient
            permissions.
-e            Displays Ethernet statistics. This may be combined with the
            -s
            option.
-n          Displays addresses and port numbers in numerical form.
-o          Displays the owning process ID associated with each connection.
-p proto    Shows connections for the protocol specified by proto; proto
        may be any of: TCP, UDP, TCPv6, or UDPv6.  If used with 
            the -s option to display per-protocol statistics, proto may be
            any of:
        IP, IPv6, ICMP, ICMPv6, TCP, TCPv6, UDP, or UDPv6.
-r          Displays the routing table.
-s          Displays per-protocol statistics.  By default, statistics are
            shown for IP, IPv6, ICMP, ICMPv6, TCP, TCPv6, UDP, and UDPv6;
            the -p option may be used to specify a subset of the default.
-v          When used in conjunction with -b, will display sequence of
            components involved in creating the connection or listening
            port for all executables.
interval    Redisplays selected statistics, pausing interval seconds
            between each display.  Press CTRL+C to stop redisplaying
            statistics.  If omitted, netstat will print the current
            configuration information once.

Sysinternals TCPView

I would also recommend running Autoruns and Process Explorer, also in the Sysinternals Suite to help diagnose your problem.

If you're looking for a simple look at which connections are hungry on Windows 7 onwards, then bring up Task Manager, Performance tab, Resource Monitor, Network tab.