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My chemistry textbook says that metals form ionic or cordinate bonds whereas non metals form covalent bonds. But in another textbook I read that Lithium, Beryllium, Aluminium, Chromium, Manganese etc, which are metals, form covalent compounds like LiCl, BeH2, AlCl3, H2CrO4, H2MnO4 respectively
Now my question is whether metals form covalent bonds or not? And if yes then how to predict which metallic compound contains ionic and which contains covalent bonds?
The first thing that should be said is that there's no difference between a coordinate bond (dative bond) and an ordinary covalent bond. Yes, the electrons "come from different places"; but the molecule doesn't actually know this, nor does it care. Once a covalent bond is formed, it is a covalent bond, regardless of where the electrons "come from".
Some of the examples you have given can equally be described as coordinate bonds or covalent bonds. For example, $\ce{MnO4-}$ could be viewed as four $\ce{O^2-}$ ligands forming coordinate bonds to a central $\ce{Mn^7+}$. Or, you could also just view them as plain old Mn=O covalent bonds. It doesn't actually matter.
Having established that there is no real difference between coordinate bonds and covalent bonds, the only real question is how can you tell a covalent bond apart from an ionic bond. The answer is mostly to do with polarisability. If you imagine an ionic bond between a strongly polarising cation ($\ce{Al^3+}$) and a polarisable anion ($\ce{Cl-}$), the cation will pull over some of the anion's electron density: this is exactly what is meant by "polarising". This changes the nature of bonding from an ionic model (where the bonding electrons reside entirely on one atom) to a covalent model (where the bonding electrons are somewhere in between the two atoms). You'll notice that all the cations you've listed: $\ce{Li+}$, $\ce{Be^2+}$, $\ce{Al^3+}$, (formally) $\ce{Cr^6+}$, and $\ce{Mn^7+}$, are very highly polarising. The first three have a very small size, and the last two have exceptionally large positive charges.
That said, covalent bonds between two transition metals are reasonably common, too: a simple example is $\ce{Mn2(CO)10}$ (Wikipedia), which is essentially $\ce{(OC)5Mn-Mn(CO)5}$ with a single bond between the two.
The bond order between two metals can go up to slightly ridiculous numbers. Cr–Cr and Re–Re quadruple bonds are well-known in $\ce{Cr2(OAc)4(H2O)2}$ and $\ce{[Re2Cl8]2-}$ respectively. Even quintuple and sextuple bonds between metals have been described (although some of these accounts are not without controversy).
How can you predict this, though? It's not trivial at all, and in truth you probably don't need to care about these exotic examples until you have enough experience to understand the bonding.
$\ce{Na2}$ and $\ce{K2}$ do exist in the sodium and potassium vapors. This can be proved by analyzing the highly diluted flames produced by hot vapors of these metals in contact with diluted $\ce{Cl2}$ vapor, as proved by Polanyi and co-workers. See the following references: M. Polanyi, Atomic Reactions, Williams and Norgate, London (1932); M.G. Evans, M. Polanyi, Transactions Faraday Society 35, 178 (1935); C. E. H. Bawn, Ann. Repts. Chem. Soc. 39, 36 (1942).
Your textbook is right. Lots of metals form covalent bonds. In the case of lithium chloride such bonding is one explanation for the solubility of this compound in organic solvents (see this answer).
One additional example you might want to know about is Grignard reagents, a class of highly basic compounds in which carbon is covalently bonded to magnesium. These are widely used in organic synthesis processes.
