One of the harder things about starting a CI-oriented approach is realizing how low our own Latin proficiency is. Even famous PhDs like Mary Beard admit their reading fluency isn't what you'd expect from someone who "knows" Latin. Her conclusion, however, that even the Romans probably didn't "know" Latin is... questionable, to put it mildly. See John Kuhner's response.

Advice & Strategies

Language acquisition works the same way for you as it does for your students. That is to say, what you need is input. Lots of input.

Strategy 1: Extensive Reading

Extensive reading means reading--READING, not translating-- a LOT of Latin that's at or even below your reading level. You're looking for something you can truly read and understand in real time, without having to parse or "go to the verb" or think about uses of the ablative. 
Easy Reading Materials 
  • Your textbook: For real. Read through the passages and you will find your Latin improves. It's also nice because chances are good the vocabulary is familiar and reading it will help you teach your students better too.
  • Lingua Latina Per Se Illustrata: I can't count the number of times I've heard things along these lines from people who allegedly already knew Latin: "I read through all of Familia Romana and at the end it was like I'd really LEARNED Latin." Nor should you neglect the second book, Roma Aeterna, or other supplementary materials. The audio recordings of FR are a good way to get that input as well.
  • Studium Angelopolitanum: Beautifully written texts from Alexis Helmer, including unofficial supplements for LLPSI.
  • Tres Columnae: From Justin Schwamm et al. Connected readings about families in the Bay of Naples.
  • Reading Material in Latin: A nice list of readings by level, including some out of copyright old "easy Latin readers" that are charming in an early twentieth century sort of way. There is a lot here.
  • Narrationes Faciles de Historia Romana: compiled by John Piazza from a variety of textbooks.
  • Narrationes Faciles de Mythologia Romana Graecaque: compiled by John Piazza from a variety of textbooks.
  • Latin Novellas: The Latinity in these varies, but they are certainly easy to understand, which is mainly what you need.

Less Easy (mostly for vocab reasons) Reading Materials
  • Latinitium Blog: by Daniel Pettersson, an erudite and charming spoken Latinist, who is really good at finding interesting stuff to talk about in Latin.
  • quo diversus abis: a sort of daily-life blog from a young woman who writes beautiful Latin. Very easy to read. She also links to a variety of other Latin blogs. There are tons of these out there!
  • Medieval Latin & Christian Latin from The Latin Library:  Not all easy, but a lot of it is.
  • Risorse didattiche: Reading resources from the Accademia Vivarium Novum. A real treasure trove- tons of material here, at various levels. You do have to click around a bit in the sidebar to find them.
  • Project Arkhaia: From Kevin Ballestrini et al., a wonderful collection of tiered readings from the AP Caesar & Vergil syllabi, plus some others.
  • The Latin Vulgate: Yep, the Latin Christian Bible. Compelling for some and well known for many, and the Latin is generally considered easier than earlier classical texts. oh and also probably the most popular and widely used Latin text of all time... no big deal.
  • Vicipaedia: Read in relatively simple Latin about a huge variety of things.
  • Ephemeris: A daily(?) newspaper in Latin about everything from current events to... this guy. Huge archives. Includes other stuff too like Sherlock Holmes stories.

Strategy 2: Extensive Listening

Same concept as above, but for audio. 
And here are some that are videos.
  • Latin Media from Latinitium: Videos and audio of various people speaking, teaching, and lecturing in Latin, as well as music & poetry. 
  • 50 Hours of Spoken Latin: A collection of audio and video from a lot of different sources on YouTube, brought to you by Daniel P. of Latinitium.

Strategy 3: Learn to speak Latin

It's a little bit of a hot potato to say output is necessary for language learning, and I'm not interested enough in theory to get into it. What I am going to say though is that learning to speak Latin has certainly improved my ability to read it, if only because it puts me in situations where I have to negotiate meaning in real time with other Latin-speakers. So head on over to the Spoken Latin section and dip a toe in, particularly in the Why speak Latin? section. It's a wonderful, weird world.