What metrics will you look at in order to see the project is moving successfully?

Most metric sets deal with a variation of these attributes and are chosen to help project managers gain insight into their product (size, software quality, rework), process (rework, software quality) and project (effort, schedule).
But below is a broader classification :
Project Management Metrics
milestone metrics

number of milestones
number of proved requirements per milestone
controlling level metrics risk metrics
probability of resources availability
probability of the requirements validity
risk indicators (long schedules, inadequate cost estimating, excessive paperwork, error-prone modules, canceled projects, excessive schedule pressure, low quality, cost overruns, creeping user requirements, excessive time to market, unused or unusable software, unanticipated acceptance criteria, hidden errors)

application risk metrics workflow metrics
walkthrough metrics
traceability metrics
variance metrics controlling metrics

size of control elements
structure of control elements
documentation level
tool application level management database metrics

data quality metrics
management data complexity
data handling level (performance metrics)
visualization level
safety and security metrics Quality Management Metrics
customer satisfaction metrics

characteristics size metrics
characteristics structure metrics
empirical evaluation metrics
data presentation metrics review metrics
number of reviews in the process
review level metrics
review dependence metrics
review structure metrics
review resources metrics productivity metrics

actual vs. planned metrics
performance metrics
productivity vs. quality metrics efficiency metrics
time behavior metrics
resources behavior metrics
actual vs. Planned metrics quality assurance metrics
quality evaluation metrics
error prevention metrics
measurement level
data analysis metrics Configuration Management Metrics
change control metrics

size of change
dependencies of changes
change interval metrics
revisions metrics version control metrics

number of versions
number of versions per customer
version differences metrics
releases metrics (version architecture)
data handling level