lymphocytes are able to recognise whether an antigen on a cell is foreign or not. When it comes across a foreign antigen, they start producing antibodies which are y-shaped proteins that have a specific binding site for a particular antigen. It can take a while for the lymphocytes to produce the correct antibody for the antigen so while this is happening you may start to feel ill. Once the correct antibody has been developed the lymphocytes start producing loads of the antibodies by dividing and attach them to the foreign antigens of the pathogens. This causes the pathogens to clump together which makes it easier for white blood cells to find them and engulf them.
Your explanation is correct, apart from the part about taking time to produce the correct antibody - the specificity of the lymphocytes somewhat guarantee that the correct one is produced. It is a matter of quantity and allowing your body to fight it off that might take time and cause symptoms. Correct me if I've misunderstood you. Bear in mind that immune response itself causes inflammation and fever.
Moreover, other cells known as Antigen presenting cells will eat the foreign cells by phagocytosis and present these antigens as well, which will be recognized by lymphocytes and activate them. In summary, lymphocytes could self-activate when meeting antigens, or be activated by other cells presenting antigens to produce antibodies.
Let me know if you have any other questions.
I would like to add a couple of points, if I may:-
1. It is useful to know that the cells that synthesize antibodies are the plasma cells, which are modified B lymphocytes [the latter become plasma cells on encountering antigen by developing extensive RER and mitochondria, both necessary for protein synthesis [the latter cos energy is needed for the creation of chemical bonds]. Phagocytosis is performed by three classes of leucocytes [Greek leucos = white e.g, leucoderma = a disease in which the skin loses all melanin in parts giving white patches [more visible in coloured people obviously] cytos = container here cell - similarly erythrocyte = red cell e.g. erythema = red rash on skin], namely neutrophils [so called as they are stained by neutral dyes [pH wise]], macrophages and T lymphocytes [killer T cells].
2. There are five main classes of antibody, named after which category of heavy chain they have [an antibody molecule consists of two heavy chains and two light chains joined in a part-double Y shape by disulphide bridges e.g. IgA is present in secretions e.g. tears and breast milk; IgG is the most abundant antibody ["Ig" stands for immunoglobulin, also called gamma-globulin [one sub-category of the plasma protein globulin; another is alpha-2 globulin e.g. ceruloplasmin, which binds copper]].
3. Antibodies do make itv easier for phagocytes to ingest antigens/pathogens [called opsonization], but they also destroy antigens by several other mechanisms e.g. precipitation, aggregation.
4. There is another substance called complement [a group of chemicals], which aids both B and T lymphocytes.
Don't worry if you can't remember all this detail - I teach my students more than what is required, so that what is required becomes a trifle [yummy - ] - also detail outside the syllabus can earn you extra marks in the synopsis Q.
M [specialist biology tutor]