I know in theory what rhythmic values are such as quarter note, half note, and whole note etc.., and I can do linear scale exercises very quickly without metronome, but I feel difficult to integrate them into my practice routine when using the metronome. I know that to be a better musician we always need to use a metronome when practicing and playing but how should I measure my timing? I start with the lowest speed at metronome but I get lost afterwards. Are there any tips you can suggest or how to cope with this situation in an efficient way?
Set the metronome to a bpm that you fell comfortable. Try 70 bpm for instance. For starters, consider each beat of the metronome as a quarter note. So, if you want to play a quarter, it will last as long as one beat. If you want to play a half, the note will last as two beats; if you want to play eighths, the note will last half a beat, meaning that there are two halves in each beat.
Here is a good image that explains the relations between the values:
I teach some really young kids drum, so we've spent a fair amount of time on the very basics.
Make sure you know the notes and how they relate. I have each student make a set of flash cards, and then I pick two at random. First, we start with "larger" and "smaller". Then we go to level two - how many of the smaller go into the larger ("There are two eighths in a quarter"). I start with notes right next to each other (1 step larger or smaller), but eventually we get to trickier ones ("There are 8 thirty-seconds in an eighth"). Never say 6. It's not the right answer until you add dotted notes.
Using the flash cards and a metronome set slow (60-70ish) I hold up or say the names of different notes and have them play that division (assume the metronome is clicking quarters). So I'll say "Half notes" and after a few measures, say "sixteenth notes". Again, start with just one step larger or smaller, since those are easy to feel. Later add larger jumps. You can do this with anyone, they don't have to know anything about music. You'll be able to tell if you're right or wrong.
Learn a counting system. I've used and like both 1-ta-te-ta and 1-e-and-a.
Start sightreading with really simple, short examples (8ths and 4ers, 4 measures only). No syncopation. Try to read whole pages with breaks every four measures, and don't worry about pitch. Just tap on your knee and/or count aloud. Stay slow, and you might find that initially having someone click or tap steadily will be more forgiving than the austere metronome. Add more durations only when it's easy to do the others. Some people take 1 week to get all the "normal" durations (whole - 32nds), while others take a month or more. ALWAYS tap your toe inside your shoe. It seems stupid, but performance happens without a metronome, and that foot tap can help keep your nerves in check to keep the speed from getting beyond the technique.
Now start trying to sight read with a metronome. Again, go slow, but not too slow - too slow can be harder than too fast. Keep it short and significantly easier than your technique allows.
Anytime you come to a difficult passage, first work it out by tapping or counting aloud, then add the pitches and technique, working up to the right tempo.
Once all the normal durations are easy, start doing exercises in #4 with dots and ties and syncopation. Once you can do them without melody, do them with pitches and melodies. In case you missed it above, tap your toe.
Usually the answer is to slow down and focus on a sense of physical proficiency.
Find the speed at which you can play along, then slightly increase to the point at which you have only slight difficulty. Improving may take time, so I recommend just accepting that and being patient.
Try repeating phrases, perhaps taken from a larger piece you're playing so that you may more successfully chain them together later to reconstitute the piece.
A lot of musicians are impatient and don't spend enough time focusing on aspects like their ability to match tempo or control expressiveness. That's up to them, but those skills give you more power over your music. Being able to match tempo is an important goal but I'm not implying that people should always play in strict tempo. They should just develop the ability to choose whether they're ahead, behind, or on the beat.
Recording yourself with audio and MIDI are very useful if you truly want to learn about your timing. MIDI (if available to you) is a great way to see, in numbers, where your notes are really starting and stopping (the end of a note is important, too). MIDI can also point out inconsistencies in dynamics, to boot.