How can I enable font smoothing when I connect to a Windows server with Terminal Services Client?

Use one of the following experience options:

rdesktop -x 0x8F mywinserver   # modem default + font smoothing
rdesktop -x 0x81 mywinserver   # broadband default + font smoothing
rdesktop -x 0x80 mywinserver   # LAN default + font smoothing

From http://katastrophos.net/andre/blog/2008/03/10/rdesktop-connect-to-windows-vista-with-cleartype-font-smoothing-enabled/

Fonts will be better

Using remmina one can also enable font smoothing.

TL;DR : Choose the Best (slowest) "Quality" setting in the Advanced Tab when editing the RDP connection.

Longer answer:

When editing an RDP connection, there is a "Quality" setting under the Advanced tab. It can have one of the values Poor, Medium, Good or Best. Under Prefrences/RDP one can configure properties for each of these Quality settings like this:

By default only Best has "Font smoothing" enabled, so choose Best or configure the Quality setting you want to use to include Font smoothing.

(Testing using remmina version 1.2.0-rcgit-26 (git rcgit-26) from a ppa on Ubuntu 14.04)

Perhaps not completely helpful, since Wishi has already suggested it, but for the record, there's no reason that Remmina should be displaying "terrible" fonts. You won't get Cleartype, sure, but they should still be slick and readable.

Here's a screenshot from my Windows 2003 box :

That's running 16-bit colour, though. In order to try that, go to the settings page in Remmina and change it from 256 colours to 16-bit :