This is a classical protein purification problem - you have to find ways to fractionate your mixture so that each fraction can be assayed for the activity you are interested in. When you find the active fraction you then subject that to a different type of fractionation.
The solubility of proteins is affected by the ionic strength of the buffer. If the ionic strength is raised above a critical value (which is a characteristic of an individual protein) the protein will precpitate, usually without being denatured. By doing a series of such precipitations at increasing ionic strengths a protein mixture can be fractionated according to this property. The salt used is ammonium sulphate, because of its very high solubility. You can find more details at this Wikipedia page (ammonium sulphate precipitation).
Ion exchange chromatography
Proteins have different intrinsic charges, and the net charge on a protein will vary with pH. This property is exploited in ion-exchange chromatography. In this technique the protein mixture interacts with a solid matrix carrying fixed charges. Under the correct conditions your protein plus others will bind to the matrix, while lots of other proteins won't. That's already a purification step, but you can then treat the matrix with buffers of increasing ionic strength. Weakly-bound proteins will be eluted from the matrix first, and as the ionic strength of the elution buffer increases proteins will elute in order of strength of interaction. By performing this procedure with the matrix in the form of a chromatography column and collecting eluate fractions you can achieve another step in the purification of your protein.
Size exclusion chromatography (gel filtration)
This technique is used to fractionate a mixture of molecules according to size. In this case the solid matrix that is used is charaterised by pores of a specific size. Proteins that are too large to enter the pores will pass through quickly; small proteins will spend a lot of time moving in and out of the pores and so will pass through slowly; and proteins of intermediate size will spend some time in the pores. Essentially the average time spent passing through the matrix will depend upon size. Read more here.
These are just three of the possible fractionation procedures - others, including hydrophobic interaction chromatography are described here). The key point to grasp is that using any one of these techniques will give you your protein in a fraction which also contains other proteins, but if you apply them sequentially you will progressively remove contaminants because each approach is based on a different physical property of proteins, and your target will have a unique combination of these properties.
At each step you will be assaying fractions to locate the protein of interest, and also running fractions on SDS-PAGE to get a feeling for the number of proteins left. If you have an assay that lends itself to measuring some kind of unit of activity (as you would with an enzyme), then by also measuring the protein content of the active fraction you can express the purity as a specific activity (units of activity per mg protein). At each step in your purification this specific activity should increase because you recover most of the activity, but discard a lot of the proteins that were present in the previous material. The increase in specific activity can be used to assign
a purification factor to each step. A useful purification step will give a reasonable yield in combination with sigificant purification.
Finally, what order to do these steps in? I've listed them in a logical order - salt precipitation is a low-tech method that can give a quick first step and which has the advantage that it concentrates the target protein (since the precipitate is redissolved in fresh buffer). Size-exclusion chromatography doesn't deal well with large amounts of protein so is best left until later. However, protein purification is a craft, and an expert will use the available techniques according to the exact nature of the project.
From the tone of your original question I had inferred that you were a beginner, but from your comments, apparently not, since you are thinking in terms of mass spec as an endpoint. A starting concentration of pg/mL is quite a challenge, and clearly you are never going to have enough material to see your molecule on standard gels.
Here are a couple of thoughts on aspects of your problem.
It seems that in trying to purify minor components of serum a good starting point is to deplete the serum of its major proteins. This is a link to an imunodepletion product, and here is a related product from Biorad (no connection with either company and no recommendation implied).
I also found these papers which may be of interest. (Incidentally I think you'll find that each of the techniques I described is still used, perhaps just with modern instrumentation.)
Li,J et al. (2012) Purification, identification and profiling of serum amyloid A proteins from sera of advanced-stage cancer patients. JOURNAL OF CHROMATOGRAPHY B-ANALYTICAL TECHNOLOGIES IN THE BIOMEDICAL AND LIFE SCIENCES 889: 3-9
Fertin, M et al. (2011) Strategy for purification and mass spectrometry identification of SELDI peaks corresponding to low-abundance plasma and serum proteins. JOURNAL OF PROTEOMICS 74: 420-430
Finally, a quick look at the Wikipedia page on plasma reveals that brain natriuretic peptide seems to be present at the levels that you are considering perhaps you should look for papers describing its purification.
Other than that I think you have to hope that a proteomics expert will come by and give you an expert answer.