*Please note that having excessive tension in the hands will prevent you from playing in tune no matter how good your ear is! Developing a healthy technique is vital to be able to play in tune.
Ringing tones are the first thing to learn and practice to help stabilize your intonation.
On the cello, the ringing tones are every A, D, G or C that you play with a left hand finger (not the open strings).
Play a D on the A string in first position and then play the open D. Try to make this octave sound in tune by moving all the fingers (and thumb) higher or lower in pitch. (Don't move only the 4th finger or you'll put tension in the hand) An octave is an easy interval to notice whether or not it is in tune. (When you try this, don't just move your pinky higher or lower to get the high D in tune. Loosen your thumb and let all the fingers slide up and down the string until you find the pitch that sounds an octave above the open D)
If you look closely at the D string, you might notice that when playing the high D in tune, the open D is vibrating. If you try this with a C, you'll definitely be able to see the C string vibrating since it's a thick string. This is very exciting and it's called a sympathetic vibration.
Improving your intonation
When you are trying to improve the intonation of a phrase, you must play each note long enough to give your brain enough input to register the pitch. Students often think they are in tune even though they are not in tune because they are playing too fast. They think the phrase sounds OK but each pitch is not registering long enough for the brain to make a correct judgement.
Leading tones - I teach students about leading tones as soon as they can play a scale. If they are used to the sound of a piano's half steps, they are in for a surprise. The easiest way to understand what a leading tone is, is to take the 7th note of a scale and play it sharp. This note "leads" to the 8th note of the scale which is also the first note of the next higher octave. The amazing thing about this, is that the faster the tempo, the sharper the pitch of the leading tone must be, in order to have beautiful melodic intonation. The half steps on a piano or guitar are all more or less the same but on the cello (or in singing) we can alter the half steps to produce beautiful effects. Amazingly, the pitch of leading tones is dependent on the speed of the notes in the phrase.
Major/minor
When you play the 3rd note of a scale, play it on the sharp side for a major scale and on the flat side for a minor scale. You can over do this so listen carefully and decide how much you dare to raise or lower this crucial note which gives so much character to the scale or to the key you are playing in. Again, the pitch is dependent on the speed of the notes so you have to use good taste - something you will develop by listening to fine string quartets or string ensembles and choirs.
Double Stops - When you are ready to learn Double stops, you must learn a different type of intonation. Go to Double Stops for more detail.
"Perfectly in tune" in an expression I only use for double stop intonation when playing unison, octave, P4 and P5. I use the expression "beautifully in tune" when referring to other intervals or melodic phrases since intonation is subjective to a certain extent.
Listen to professional string quartets to hear how their intonation suits the music they play. They use "Just" intonation to play chords in tune in Classical Period Music. Intonation is a fascinating topic and can get complicated. I highly recommend the book CelloMind
Electronic Tuners - These are great for students to use to fine tune their cellos. They should not be used to see if pitches in your music are in tune. This is because intonation is much more complicated than using the pitches that the electronic tuner says is in tune. Just use it to tune your open strings. A simple electronic tuner like this Korg is good.