My experience with cell cultures may not be that extensive compared to some others here. But I have personally had an experience with mammalian primary cell cultures, mouse and human cell line cultures, and microbiology cultures, so I am glad to share my experience with you.
1. You need to add the correct nutrients needed by the cells
When you order cells from ATCC (American Type Culture Collection), they would give you a manual on what is the appropriate medium for the cells. For mammalian cells, for instance, there are RPMI (Rosewell Park Memorial Institute) 1640, EMEM (Eagle's Minimum Essential Medium), DMEM (Dubelco's Modified Eagle's Medium); some media contain L-glutamine and some do not; some contain phenol red (as pH change indicator) some do not. Currently, there are endless list of culture media's supplements commercially available. Just remember that different cells may grow differently in different media.
2. You need to use the correct vessel
Some cells can only grow in hydrophilic surface, such as gamma ray-treated polypropylene T-flask that is commonly used to grow mammalian cells. The hydrophilic surface is needed for some cell type before they can even propagate and divide. However, if you are culturing a mouse macrophage cell line, RAW264.7, it would require harsh scraping or trypsinization if you grow the cells on hydrophilic surface. You would need an untreated polyethylene surface that is relatively hydrophobic.
3. You need to seed at appropriate cell density
Too low cell density may cause your cells taking too long to reach 90-100% confluency. Sometimes, the cells may even die when you seed too sparsely because cells need some inter-cell communication/signaling to survive and propagate.
4. You need to keep track the passage number of the cells grown
Higher passage number cells may harbour different set of genotypes compared to their parent cells (P0 or P1). As the passage number increases, cells may also lose their ability to propagate healthily. If you are comparing an effect of a variable on certain cells, usually researchers would recommend to use the same passage number of cells to ensure that the response observed is unaffected by the cells' genotype.
I would suggest others to add more fundamental points in cell culture to share!
I completely agree with Kemas Aurino's points. Adding to the answer:
1. Practice proper aseptic technique: This is not only limited to keeping your cultures safe from contamination by microorganisms, but also from cross-contamination from improper handling of apparatuses inside a cell culture hood.
2. Cryopreserve when required: Cryopreservation of cells do not only reduce the risk of microbial and cross-contamination, it is also practical especially when your cells grow at a faster rate, as you may be left with more cells in culture vessels than what you actually need.
3. Get acquainted with your cell type: Familiarize yourself with not only the general traits of normal healthy cell appearances but also specific traits which belong to the cell line you are culturing. Not all cells behave and look the same. For instance, some cell types generally possess more vacuoles (which normally indicates senescence) and can still grow healthily.