Ok - if you can get the BDNF clone from a company - this will solve some of your problems. Make sure that when you get the plasmid that you transform it into some bacteria and keep a glycerol stock or two frozen down. This will ensure that you have a source of the clone for years to come. It also pays to sequence the insert on any plasmid coming into the lab just to ensure that it is what the people who sent it to you say it is.
Depending on which plasmid the clone is in, you have two choices:
1) If it is in a mammalian expression vector (will probably contain a viral promoter such as CMV or SV-40, or perhaps a mammalian promoter), you should be able to just transfect this plasmid into the cells, and this will hopefully give you expression straight off.
2) If the BDNF is in a vector used for propagation (e.g. one of the pUC series), then you will need to subclone it into an expression vector before you can transfect and get expression. There are a number of options here, depending on the vector that you want to insert it into. The classic way to do this is to identify some restriction sites around your BDNF sequence that are also present on the expression vector (choose non-compatible ends if possible, it makes things easier), but make sure that these sites are also not present inside your BDNF sequence. You would then digest both the expression vector and the BDNF plasmid with these restriction enzymes and purify the resulting fragments leaving you with linear expression vector and your BDNF clone sequence (usually referred to as an "insert"). Then you would ligate the insert into the expression vector. Another option is to amplify the BDNF sequence with primers that contain the restriction sites you want, digest the resulting product and the expression vector and ligate.
If you don't have the clone then you can extract RNA from cells that do express BDNF, convert this to cDNA, then amplify as above with primers containing restriction sites, digest, ligate. You could also do topo-TA cloning here, which is faster but much more expensive.
Your local university library will very likely have a copy of Sambrook et al., Molecular cloning: a laboratory manual, which will be a very useful resource for you.