Programme:

Lectures 1,2:
Standard facts that can be found in any book on algebraic geometry:

Affine algebraic sets, ideals of subsets, correspondence between
ideals and algebraic sets, Hilbert's Nullstellensatz, 
morphisms of affine algebraic sets, the affine coordinate ring,
equivalence of categories of affine algebraic sets and reduced finitely 
generated algebras

Lecture 3:
Presheaves and sheaves, regular functions,
structure sheaf on affine irreducible subset

Lecture 4:
Regular functions on open subsets, germs of sheaves of regular function,
structure sheaf on spectrum of a ring

Lecture 5:
A space with functions over a field, morphisms, affine variety as a space 
with functions, prevariety.
Morphism of presheaves, morphisms of ringed and locally ringed spaces,
affine schemes, morphisms of affine schemes,

Lecture 6:
Anti-equivalence of categories of rings and affine schemes.
Projective varieties, homogeneous coordinate ring, rings and modules 
with gradation

Lecture 7:
Localization of rings with gradation, structure sheaf of a projective variety.
The only global functions on a projective variety are constant.
Finite algebras, finite morphisms, Nakayama's lemma.
Finite morphisms are surjective.

Lecture 8:
Finite morphism is closed. Chevalley's theorem.
Theorem about extending homomorphisms.

Lecture 9:
Every involution of affine space has a fixed point.
Products of varieties. Gluing schemes.
Complete varieties. Every projective variety is complete.

Lecture 10:
Finiteness is a local property.
Projections from linear subspaces. Blow ups.
Projections of varieties onto projective spaces.
Noether's normalization theorem.

Lecture 11:
Krull dimension of a ring and of a topological space. Basic properties.
Approach to dimension via transcendence degree.
Tangent spaces to hypersurfaces and affine subschemes of an affine space.
Approach to dimension via tangent spaces.

Lecture 12:
Independence of the tangent space from the embedding. Tangent space to 
a k-scheme.  
Regular rings. Smoothness is a local property and a variety is smooth if 
the local ring is regular.
Krull's theorem about principal ideals: geometric case.
Dimensions of fibres of morphisms.

Lecture 13:
Upper semicontinuity of dimensions of fibres of morphisms.
Bertini's theorem.
Grassmannians as projective varieties and Plucker embedding.
Lines on surfaces in 3-dimensional projective space.

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Suggested reading:

Gortz-Wedhorn "Algebraic Geometry I":  
Chapters 1,2,3,4 and 5  
Hartshorne "Algebraic Geometry":
Chapter 1: 1.1-1.5
Chapter 2: 2.1 and 2.2, 2.8: Theorem 8.18
Shafarevich "Basic Algebraic Geometry 1":
Chapter I
Chapter II: II.1, II.3, II.4.1-II.4.3
Reid "Undergraduate algebraic geometry"

In Polish:
A. Bialynicki-Birula "Wyklady z geometrii algebraicznej"
Rozdzialy 1, 2, 3, 5.1, 6, 7.4-7.6